Activate Yeast: In a large measuring cup, combine 3/4 cup warm milk, 1 1/2 tsp dry yeast, and 1 tbsp sugar. Whisk lightly and let sit for 10 minutes until frothy.

Mix Dough: Attach the dough hook to a stand mixer. Add 4 cups all-purpose flour, 1/4 cup sugar, 1 tsp salt, 3 eggs, and the yeast mixture. Mix on low speed, then increase to medium-high until a sticky dough forms (about 4-5 minutes).

Incorporate Butter: Gradually add 1/2 cup softened butter (cut into chunks) to the dough while mixing. Continue kneading for 5-10 minutes until the dough is smooth, elastic, and no longer sticks to the bowl.

First Rise: Transfer the dough to a lightly greased bowl, cover with a towel or plastic wrap, and let rise for 1 hour or until doubled in size.

Shape Rolls: Punch down the dough and divide it into 12 equal portions. Shape each portion into a round ball and place on a parchment-lined baking sheet or in a greased 9x13-inch baking dish. Cover and let rise for another hour.

Brush and Bake: Preheat the oven to 350°F. Prepare the egg wash by whisking 1 egg with 1 tsp milk. Brush the egg wash over the rolls and bake for 20-25 minutes until golden brown.

Cool and Serve: Allow the rolls to cool slightly before serving.