A Joseon Empire won't work unless:
A: Admiral Yi survives the Assassination attempt(possible), completes the conquest of Kyushu and the Ryukyus(also possible, but barely) and returns to confront King Seonjo and overthrow him, (No chance in hell, he was basically the Bellisarius of Korea) and re-forms the Hwarang order and invites Matteo Ricci or any of the other Jesuit scolar-missionaries (and pays close attention when they talk about administration or show of a useful Western invention) or
B: The legendary Dutch (or maybe he was Scottish or English) sailor who was responsible for sword forms 3 through 6 in the Dan Do section of the Muye Dobo Tongji takes up King Minjo's offer to help build up and train a modern Western army and Navy. He then settles down, marries a descendant of the Koryo or Koguryo dynasty (to give his children a descent from Dangun Wangeon), Destroys the Manchu invasion of Korea, leads a coup against the king when he sends him the yellow scarf when the king thinks the sailor is getting too popular and powerful. With the Manchus out of the picture for the time being, he's in a perfect position to capitalize on Li Zicheng.
Anything else would be ASB due to the cultural disdain King Saejo created for the millitary in order to prevent his overthrow the way he overthrew the Koryo dynasty. Korea simply doen't have the numbers to overcome millitary and political institutional weakness.
I say one of these PODs:
1. In the late fourth century B.C., Gojeoson manages to force a Status Quo Ante Bellum in it's war against the State of Yan. (Actually trouncing them would have woken the sleeping giant: even Chin and Chu would have rallied around the Zhou Emperor against an obvious powerful foreginer who dared to strike into the Middle Kingdom like that.) Subsequent Gojeoson kings introduce millitary reforms around Korea's greatest strength; mounted archery. (Like the stirrup, and bodkin arrows to pierce Chinese "Shan" scale armor) So that Korean forces run circles around (Liu) Han Wudi's Ji and sword squares and chariots. If they can win, or even make the victory Pyrric enough, the Xiong-Nu, Hoparu-Ho, and Donghu will break the borders and cause the Chinese Dark Ages three centuries early.
2. The Kingdom of Buryeo doesn't shatter into warring statlets. As Soon as Gongsun Yan or Gongsun Ren starts either one's rebellion, Buryeo can strike straight at the Fifty Commandries and destroy them. And since Cao Pi isn't the paragon of honor his fater actually was and the Lius of Shu made themselves to be, Korea won't be re-incorporated into the Middle Kingdom anytime soon. Then, when the Toba Turks invade the North of China...