Combine milk, yeast, and sugar in a stand mixer using a dough hook or paddle on low speed for 30 seconds. Cover with a clean cotton towel and let rest for 5 minutes.
With the mixer on low speed, add 1 egg, 5 tablespoons of butter, and 1 cup of flour. Scrape down the sides of the bowl and add the remaining flour on medium speed. Continue mixing for 2 minutes until the dough pulls away from the sides of the bowl. If the dough is still very sticky to the touch, add a tablespoon of flour at a time until the desired constancy is achieved.
Once the desired constancy is achieved, continue mixing on medium speed for an additional 2 minutes to knead the dough.
Use kitchen spray or oil to coat a bowl. Add dough, spray, or oil the top of the dough to prevent sticking and drying out. Cover the bowl with a clean cotton towel or paper towel and let the dough rest on your countertop for 1-2 hours, or until the dough has doubled in size. The amount of time will depend on the temperature of your kitchen.
Generously grease a 9×12-inch pan with 2 tablespoons of butter. Cut dough into 15 pieces using a pastry cutter or knife. Form each piece into a ball by pulling the sides towards the back and pinching together on the backside of each roll to create a smooth top. Place rolls into the prepared baking dish. Create 5 rows, with 3 rolls in each row.
Cover with a clean cotton towel, and let rolls rise until puffy, about 1 hour.
Preheat oven to 350 degrees.
Lightly beat the remaining egg in a small bowl. Brush the tops of rolls with egg to promote a golden top.
Place rolls on a middle rack in the oven and bake for 20-25 minutes. If rolls begin to brown too much, lay a piece of foil on top while baking.
Melt the remaining 3 tablespoons of butter. Once rolls come out of the oven, brush the tops with butter. Serve warm or at room temperature. Cover leftover rolls tightly and store them at room temperature for 2-3 days or in the refrigerator for up to 1 week.
The directions call for a stand mixer, the preferred method. However, you can also mix the dough by hand using a sturdy wooden or metal spoon.