Add the warm milk, 2 tablespoons of the sugar, and yeast to the bowl of a stand mixer. Let sit for 5 minutes, until it is foamy on top and you can see the yeast is active.
Add in the remaining sugar and the eggs to the bowl of the stand mixer and mix on medium low speed to combine.
Add in the flour and mix on medium low speed until a dry, shaggy dough starts to come together, about 3-4 minutes.
Add in the room temperature butter, all at once, and continue mixing until the butter is completely incorporated, about 5 minutes, then continue mixing until a smooth dough forms, at least 5 more minutes.
Place the smooth dough ball in a clean bowl, greased with the reserved 2 tsp of oil or oil spray. Cover the bowl with cling wrap and let rise until doubled in size, about 2 hours. Make the filling while the dough rises.