Combine milk and yeast in the bowl of a standing electric mixer. Let stand until yeast is dissolved, about 5 minutes. In a small bowl, lightly beat 3 of the eggs. Set aside.
Add all-purpose flour, beaten eggs, sugar, and salt to the yeast mixture. Mix on low speed until evenly combined.
Add bread flour, 1/2 cup at a time, and mix until well blended. Switch to the dough hook and knead on low speed until the dough for 8 to 10 minutes. The dough should be smooth (but still sticky) and clear the sides of the bowl.
Add butter one tablespoon at a time, vigorously kneading with the mixer until all the butter is completely incorporated and the dough becomes smooth once again.
Place the dough in a large, generously buttered bowl, cover with plastic wrap, and let it rise at room temperature until doubled in volume, about 1½ hours.
Turn out the dough, punch it down, and knead it briefly, using a small amount of flour to prevent sticking if needed. Return the dough to the re-buttered bowl and refrigerate, covered, for 4 to 12 hours.
Gently flatten the cold dough on surface dusted only very lightly with flour. Using a sharp knife or bench scraper, divide the dough evenly into 8 pieces for hamburger buns or 16 for slider buns. Working one at a time, knead a piece of dough briefly and then roll it into a ball by swirling it under a cupped hand on an unfloured surface.
Once it's shaped into a smooth ball, tuck it under your thumb and gently drag the ball across the work surface, allowing the dough to pull tightly as it smooths the ball's surface. Place finished balls on baking sheets lined with greased parchment paper. Repeat with remaining dough.
With your fingertips, gently press the balls flat. Brush their surfaces with the remaining egg lightly beaten with 1 tablespoon water. Cover with greased plastic wrap and let rise at room temperature until doubled, about 1½ to 2 hours.
Just before baking, brush buns again with the egg mixture. Bake in an oven preheated to 375° F until golden brown, about 20 to 22 minutes for hamburger buns, 18 to 20 minutes for slider buns. Switch racks midway though time through for even baking.
Cool on a rack before slicing, toasting, and serving.