Start preheating the oven to 400ºF (204ºC). Lightly butter a 9x13 baking pan.
In a large mixing bowl, whisk together the flour (3 3/4 cups), instant yeast (2 1/2 teaspoons), sugar (3 tablespoons + 1 teaspoon), and salt (2 teaspoons).
In a separate bowl or large liquid measuring cup, whisk the warm milk (1 cup + 1 tablespoon), melted butter (5 tablespoons + 2 teaspoons), and egg (1) until smooth.
Pour the wet ingredients into the dry and mix until a shaggy dough forms. Knead by hand or with a dough hook for 5–7 minutes, until the dough is soft and elastic. It may still be slightly tacky, but should pull away from the sides of the bowl cleanly.
Remove the dough onto a lightly floured surface. Divide the dough into 18 equal portions and roll each into a smooth ball. Arrange the rolls in the greased 9x13 pan.
Cover the pan loosely with plastic wrap or a clean kitchen towel and let rise in a warm, draft-free spot (about 75-80) for 20–25 minutes, or until noticeably puffy.
Preheat the oven to 400°F (204ºC) while the rolls are rising if you oven heats quickly.
Once the rolls have risen, remove the covering, brush with beaten egg, and bake for 12-15 minutes, or until golden brown and baked through. If needed, tent loosely with foil during the last few minutes to prevent overbrowning.
Brush warm rolls with additional melted butter before serving.