Moon Embracing the Sun fans, please watch The Princess Man. Both dramas came out around the same time period (late 2011-early 2012). Both are also about romance and love stories between not just one but at least 2 couples.

I know the dramas are kinda random, but I just want more people to discover more of Moon Chae Won's works. The few similarities of both stories include a love story where the male leads have great chemistry with Moon Chae Won, a couple whose love must face a lot of difficult challenges, one of the main leads lies to the other about his/her real identity which causes problems later. In addition, both dramas also show Moon Chae Won's character being a strong female lead, at one point we can see her saving the male lead (Bonus: the male leads in both dramas actually acted together in 2008 kdrama Iljimae, so watch that if you want to see both guys act together). I find it also interesting that the male leads in the drama also have good friends who are both very persistent in loving their own respective love interests who are reluctant in accepting them at first
For Flower of Evil Fans -> you can see in Princess Man Moon Chae Won's top notch acting, Princess Man should be the first of her other dramas you watch since she won or is nominated for more awards for this show than any of her dramas as leads (including 2012's Innocent Man).
For Princess Man Fans -> as you've seen Moon Chae Won's great acting already, you can see in Flower of Evil her most recent work (2020). It's the drama that is worthy of her acting skills and her first and most acclaimed drama after 2013.
Trivia: Producer Yoo Sang Won is a producer involved in both dramas. He also did Lee Joon Gi's 2014 drama Gunman in Joseon

They have the same writers. They main character is also a twin in both dramas. There is a prophecy about the twin characters in the drama that is important to the plot. They also both have complex characters, meaning likeable villains, protagonists with a dark side, and characters which we won't understand at first if they're good or evil

Both based on the same novel, both entertained the possibility of historical Korean painter Shin Yun Bok as a woman pretending to be a man because her paintings mainly depict women. The two stories interpret the story of Shin Yun Bok being a real woman quite differently, but the similarity in both shows is that the historical character Kim Hong Do(who was a fellow painter of Shin Yun Bok in Joseon era) is her teacher and is also one of the 2 characters who fell in love with her. Both shows depict Shin Yun Bok's life as a woman loved by 2 very different people while living her life as a painter.

Both are Joseon period dramas about a woman's struggles to become a great artist. I really love both dramas because both plots explain the characters very well through the use of their art forms. Both dramas also feature a beautiful and talented gisaeng played by two great actresses (though the gisaeng in Painter of the Wind is NOT the lead female). A scene I find quite similar in both dramas is both feature a scene wherein the gisaeng is playing the gayageum instrument while her lover (in Painter of the Wind, this lover is the female lead) is painting as the lover listens to her music.

Both depicts a story of a woman who was born a noble yet circumstances in her life led her to be linked with a much older man (an official who owns her as a gisaeng in Hwang Jin Yi, a professional painter who became her trainer in Portrait of a Beauty). Both women are very talented, either as a poet or as a painter. Both older men are very much attracted to the young woman linked to them. Despite the older men's attraction, both young women fell in love with a man closer to their age who basically has a low position in Joseon's class based hierarchy. The love triangle in both these shows are then lead to chaos on the three main characters' lives until the ending.

Before actor Ryu Seung Ryong became popular with Miracle Cell No. 7 and Kingdom Netflix series. I have seen his great performances as a villain in these two shows that has a gisaeng as an important character.
(Bonus: As a result, while most people are surprised that the kind but imprisoned father in Miracle Cell 7 is the same guy in Kingdom, I am actually surprised that he could pull off a kind character in Miracle Cell 7 after seeing his villainy in these 2 shows).
These are the similarities of the 2 shows:
1. Both shows have a gisaeng as an important character (though the gisaeng in Painter of the Wind is not the main character)
2. Both gisaengs are beautiful and is the most desirable gisaeng in the land
3. The main villain in both shows is a much older official played very well by actor Ryu Seung Ryong, who lusts after the gisaeng. He was able to own the gisaeng as a property.
4. Because the gisaeng loves another "man", Ryu Seung Ryong's character wants to get rid of that "man".
5. The "man" the gisaeng loves in both shows are very talented in their own field (the one in Painter of the Wind is good at painting, the one is Hwang Jin Yi is good in martial arts)

Done around the same time period, 2006-2007, both shows are different retellings of Hwang Jin Yi. If you want to focus more on Hwang Jin Yi's talent as a gisaeng, as well as colorful costumes, then you might want to watch the T.V. series. If you want to watch a depiction much closer to her real life (according to some people), you can watch the movie. It's good if you watch both though, as both shows depict two different views of the modern world on the historical Hwang Ji Ni and the impact of her charms and talents to the modern world.

Similarities:
=> Both are stories of a romance between a woman who is supposed to be a crown princess and a man destined to be king
=> A 12 year old Kim So Hyun gives a very excellent performance as the young version of the woman who usurps the crown princess position. (In a way she's actually the one who played the usurpers, not the adult actresses, because the takeover of the crown princess position from the real crown princess is done by her character in her teen years).
=> We see the male lead (or rather the young version of the male lead in Moon Sun) attracted to a girl with a butterfly in the first episode.
=> The male lead (him or his doppelganger in Rooftop Prince) has a close relative who can be a threat for high position he has (the king position in Moon Sun, being the heir of the company in Rooftop Prince)
=> Lee Tae Ri/Lee Min Ho actor born 1993 gave excellent performances in both dramas. Both his characters in a drama is a very smart young man who helps one of the main characters.

If you enjoy watching Han Ji Min falling in love with a crown prince in Rooftop Prince, Yi San is also another story where we can see her fall in love with a crown prince.
There is a great genuine chemistry between the crown prince and Han Ji Min's character despite the fact that her romance with the crown prince seems impossible.
Be mindful that Rooftop Prince fans might not like the politics of Yisan or Yisan fans might not like the modern era in Rooftop prince, but both dramas are worth watching to see a blossoming romance between a crown prince and Han Ji Min's character

Similarities
=> Both are stories of women who are considered "action" girls. They both eventually became the empress of a Chinese empire.
=> The emperor doesn't realize the girl's real identity as a woman until much later episodes. Its because he thought she was a man when he first met her
=> There's a lot of palace intrigue. The prospective bride of the emperor, basically tortures the female lead because she was jealous of her.
=> Both emperor's have a "loyal" eunuch.

Jumong might be too long for Sungkyunkwan fans, but I think its worth for new fans of Sageuk from those who watched Sungkyunkwan to try watching a historical drama that was considered one of the best in the previous decade (2000s).
Here are the similarities:
=> Jumong is one of the few dramas BEFORE Sungkyunkwan Scandal was aired that shows a "romance" between two male characters
=> The same gender relationship in Jumong is more two-sided than the one in Sungkyunkwan
=> Like Sungkyunkwan, one of the two males who might be involved in a heartwarming relationship with another man is a hairy man while the other one is a flamboyant pretty man who likes wearing bright colored clothing (the flamboyant man in Jumong being more womanly than the one in Sungkyunkwan).
=> If you are in Sungkyunkwan for strong, determined, female characters, Jumong has them too.
=> Both dramas have some episodes dedicated to martial arts competitions. Examples including archery and swordsmanship (using sword sticks)
=> Both feature an evil male character close to the age of the male leads.