• In a large bowl, combine 1 cup flour, sugar, yeast, and salt. Set aside.
• Crack the egg into a separate small bowl and beat it lightly with a fork. Set aside.
• In a small saucepan, melt 2 tablespoons butter over LOW heat. Add the milk and warm it for 1 minute or less, just until it reaches 120- to 130 degrees F. when tested with an instant read thermometer. It will reach this temperature quickly - if it surpasses it, just remove the pan from the heat and let it rest to cool down to the temperature range. Add to the dry ingredients, and stir with a wooden spoon or rubber spatula until well combined.
• Sprinkle ½ cup flour over the dough and then add the beaten egg. Stir again until combined. Then, add as much of the remaining flour as needed, in small increments, stirring in between each addition, until you have a soft, sticky dough. You may not need to use the entire amount of flour. When the dough is just beginning to hold together in a ball but is still very sticky, coat your hands with some flour and gather the dough up, scraping off the sticky dough from the bottom and sides of the bowl. Use your hands to form it into a neat ball and set it back down into the bowl.

Rest the Dough/First Rise

• Cover the bowl with a clean kitchen cloth and let the dough rest in a warm place for about 30 minutes. The first rise is more about letting the dough rest - it will rise slightly but probably less than double.

Shape the Rolls

• Coat a 9-inch round cake pan with softened butter or nonstick cooking spray.
• Using a bench scraper or sharp knife, divide the dough into four equal portions. Then, cut each portion into three equal pieces, for a total of 12. Try to keep the pieces as evenly sized as possible. Shape them into smooth balls. Place them in a circular pattern in the greased pan (9 around the outside edge, 3 in the middle), leaving a little space between each ball.

Second Rise

• Cover the pan with the kitchen cloth and let the dough rise again for an additional 30 minutes. After the this second rise, the balls of dough will have risen enough to fill the spaces between and they will be touching.

Bake

• While the rolls are rising, preheat your oven to 375 degrees F.
• Transfer the pan to the preheated oven and bake for 12 to 14 minutes or until golden brown. If the rolls are getting too brown, just tent lightly with a sheet of foil. While baking, melt the remaining 1 tablespoon of butter. Remove the rolls from the oven and immediately brush the tops with the melted butter. Serve warm.