Soften (bloom) the yeast in warm water for 5 minutes.
In a bowl of an electric mixer fitted with the dough hook attachment, combine flour, salt, sugar, dry milk powder, butter, and egg. With the motor running on low, slowly drizzle in the yeast mixture.
Increase the mixer speed to medium and mix until shiny and smooth, about 10 minutes.
Turn out the dough onto a dry surface and shape into a ball. Place into a lightly greased bowl, cover with plastic wrap, and let rise in a warm place (80 to 85 degrees, see notes) until doubled in size, about 1 hour.
Punch down the dough and allow to rest a few minutes for the gluten to relax.
Line two baking sheets with parchment paper. Turn out the dough onto a dry surface. Using a bench scraper, divide the dough into 4 equal portions and then divide those into 8 equal portions (about 1 ¼ oz each). Shape each portion into a round ball by pushing and turning the palm of your hand in a circular motion against the countertop and place 2 inches apart on prepared baking sheets.
Cover the shaped rolls with a clean kitchen towel and place the baking sheets on top of the oven. Preheat the oven to 400 degrees and proof the shaped rolls until doubled in size, about 30 minutes.
Gently brush the tops with egg wash. Bake until golden brown, about 18-20 minutes, rotating the baking sheets halfway through. Serve warm or allow to cool, place in Ziplock bags, and freeze for another night.