In a shallow bowl, whisk together the milk and eggs until well combined. Dunk both sides of the buns in the egg mixture until completely saturated, about 20 seconds on each side.
In a medium saucepan set over medium heat, melt the butter and place the buns on top. Using a spatula, lightly press down on the buns so they make contact with the hot butter. Cook until golden, about 5 minutes on each side.