In a small saucepan over medium heat, combine the milk and butter and heat until the butter is melted, about 7 minutes. Remove from the heat and let cool to 105° to 115°F Add the yeast and stir until the yeast is dissolved. Let stand for 10 minutes.
In the bowl of a stand mixer fitted with the dough hook, combine the 4 cups flour, the sugar and salt and beat on low speed until combined, about 30 seconds. Add the milk mixture and knead until the dough forms a ball, about 1 minute. Increase the speed to medium-low and knead until the dough is smooth and elastic, 4 to 5 minutes. Remove the dough from the mixer bowl, oil the inside of the bowl and return the dough to the bowl. Cover tightly with plastic wrap and let the dough rise in a warm place until doubled in volume, about 1 hour.
Line a baking sheet with parchment paper. On a lightly floured work surface, roll out the dough into a 10-by-7 1/2-inch rectangle. Using a ruler as a guide, cut the dough into 2 1/2-inch squares. Transfer to the prepared baking sheet, spacing the buns evenly apart, and cover tightly with plastic wrap. Let rise in a warm place for 30 minutes.
Preheat an oven to 400°F.
Remove the plastic wrap from the baking sheet. Brush the tops of the buns with the egg mixture and sprinkle with sesame seeds. Bake until the buns are golden and an instant-read thermometer inserted into the center of a bun registers 190°F, 14 to 16 minutes. Transfer the buns to a wire rack and let cool completely. Cut in half and use as hamburger buns. Makes 12 hamburger buns.
For slider buns: Follow the instructions above but roll out the dough into a 9-inch square. Cut into 1 1/2-inch squares and place on 2 parchment-lined baking sheets. Cover tightly with plastic wrap and let rise in a warm place for 30 minutes. Position 1 rack in the upper third of an oven and 1 rack in the lower third and preheat to 400°F Brush the tops of the buns with the egg mixture and sprinkle with sesame seeds. Bake for about 14 minutes, rotating the baking sheets from top to bottom and 180 degrees halfway through baking. Makes 36 slider buns.