Proof Yeast. In the bowl of a stand mixer, add the warm water. Sprinkle the yeast and 1 teaspoon of the sugar. Let this mixture sit for about 5 minutes, until it is foamy.
Add the Other Dough Ingredients. Add the warm milk, the eggs, salt, the remaining sugar and melted butter. Stir with a wooden spoon until the eggs are dispersed and the ingredient are incorporated.
Add 3 cups of flour, one cup at a time, stirring with the wooden spoon after each addition. Then, add the half cup of flour and stir it in. The dough should begin to form a shaggy, sticky mass.
Knead the Dough. Attach the dough hook to the stand mixer. Start to run the mixer at medium-high speed to knead the dough. Knead for 5 minutes, then start to sprinkle the remaining ½ cup of flour gradually as the mixer runs, kneading for another 7-10 minutes. The dough will first gather in the center while still attaching to the sides of the bowl. As the kneading progresses, the dough will start to release from the sides of the bowl. The gradual sprinkling of flour will greatly help in this process.
Eventually, the dough will start to slap the sides of the mixing bowl. It should be smooth, stretchy and passes the windowpane test (see note 1).
Let the Dough Rise. Gather the dough into a ball. Place it in a lightly greased bowl. Cover with a clean kitchen towel, allow it to rise until doubled, about 1 to 1 ½ hours.
Shape the Dough. Take the dough out of the bowl and gently deflate it. In a lightly floured surface, cut the dough into 4 equal portions, then cut each portions into 4 more equal portions, for a total of 16 dough portions. Shape each dough into a smooth ball. Place each portion into a parchment-lined 9x13 inch baking pan, evenly spaced from each other (see note 2).
Let the Shaped Dough Rise Again. Cover the pan loosely with a clean towel or plastic film. Allow the dough to rise for 1 to 1½ hour, they are soft, puffy and touching.
Egg Wash. Lighty beat together egg and 1 tbsp of milk. Meanwhile, preheat the oven to 350 ℉. Brush the tops of the buns with the egg wash.
Bake. Bake the buns at 350 ℉ for 15-20 minutes, or until the tops are lightly golden.