Add 3 cups of the flour to the bowl of a stand mixer along with the yeast, sugar, salt, warm milk, softened butter, and egg.
Fit the mixer with the dough hook. Mix on the lowest setting until the flour is worked into the wet ingredients, stopping to scrape the bowl when necessary. Turn the speed up to medium and continue mixing for 2 minutes.
Mix in ½ cup flour. Once incorporated, add another ½ cup flour and continue mixing on medium for another 2 minutes, until the dough comes together into a ball.
Add more of the remaining flour only as needed. The finished dough should remain soft and a little sticky while beginning to pull away from the sides of the mixing bowl.
Place the dough in a lightly greased bowl. Cover it with a clean towel or plastic wrap and leave it at room temperature to rise for 30 minutes.
Uncover the dough and gently punch it down to release the excess air. Divide and shape the dough into 24 rolls. For more evenly sized rolls, you can weigh the portions.
Arrange the rolls in a lightly greased quarter sheet pan or 9 × 13-inch baking dish. Cover again and leave them at room temperature for another 30 minutes.
Preheat the oven to 375°F. Bake for 12–15 minutes, until the rolls are golden and baked through. If the tops are browning too quickly, loosely cover them with foil.
Take the rolls out of the oven and immediately brush the tops with the 2 tablespoons melted butter. Serve warm. Once cooled, the rolls can also be stored for later.