Whisk together 2 cups of flour, the sugar, yeast, and salt in a mixing bowl. Lightly beat the eggs and add it along with the milk to the flour mixture. Using dough hooks, mix on medium speed to combine, about 6-8 minutes.
Scrape down the sides of the bowl and the dough hook, add 1 tablespoon of butter and set the mixer to medium-high. Begin to add one tablespoon of butter about 30 seconds between each piece; be sure that each piece is fully incorporated before adding the next.
Continue mixing the dough for 6-8 min until it’s stretchy and shiny. Pour out the dough on an oiled surface and knead into a ball. Place dough in a buttered bowl and cover with a towel. Allow the dough to rise for 30 min to an hour if using instant yeast or until it doubles in size.
Once risen, punch it down and pour it out once again on to a floured surface. Knead into a ball and divide evenly into 8 pieces. Shape each piece into balls. Place each bun on a parchment-lined baking sheet and cover with plastic wrap to rise again. Preheat over to 375 degrees.
Whisk the egg yolk and milk together and brush each bun being careful not to drip the egg wash into the parchment. Bake for 12-15 minutes or until golden brown. Internal temperature with a meat thermometer should read 199 degrees. Cool before serving.