I know that saying ASB is frowned upon but this is...erm...highly unlikely would be an understatement.
For one, no Korean dynasty since the Goguryeo had much of an expansionistic streak (mainly because that tended to lead to conflict with Chinese empires and there's a bit of a discrepancy in manpower and money that moxie is hard-pressed to overcome) or an exploratory streak (there were trade connections with the Indians and Persians *long* ago but no charting the unknown). Prisoners could be sent off to one of the hundreds of islands off of the Korean coastline and trade with China (you know, that thing that European empires sought for centuries and resorted to war to get on their terms) was well-established and provided most of whatever was desired. There's nothing driving Korea to Europe the way that a hunger for spices, chinaware, and tea drove Europe overseas to establish great trade routes (and empires along the way). If they want land (which they hardly did, just look at how the borders of Korea shifted over the centuries. Other than the bit on the north, there was virtually no expansion in a millennium), Manchuria, Japan, or anything east of the Hindu Kush would be a better bet than Europe.
Not to mention that bureaucracy took precedence over the military (Confucian morals plus that brush with military dictatorship that the Goryeo dynasty suffered through), so there wasn't ever much of a drive to bulk up the military, which would help with conquests. There weren't many existential threats (the Jurchen raids were irritating but endurable, the Japan invaded once in a thousand years, the Qing invaded twice, and that's the entire military history of wars on the Korean peninsula between 1392 and 1905. If no Korean dynasty ever bothered to expand beyond a few dozen kilometers off the coast, getting them to establish an overseas empire is pretty damn hard.
Might want to push it back to before 1100. After all, the earlier the better (and more you can manipulate events reasonably).