01 - In a stand mixer bowl, combine flour, sugar, yeast, and salt. Mix briefly to distribute evenly.
02 - Add eggs and lukewarm milk to the dry ingredients. Mix on low speed until a shaggy dough forms.
03 - With mixer on medium speed, add butter one cube at a time, allowing each piece to fully incorporate before adding the next. Continue kneading for 8 to 10 minutes until dough becomes smooth, elastic, and pulls away from the bowl sides.
04 - Transfer dough to a lightly greased bowl, turning once to coat all surfaces. Cover with plastic wrap or damp towel and let rise in a warm, draft-free area for 1 to 1.5 hours until doubled in volume.
05 - Punch down risen dough to release air. Divide into 12 equal portions and shape each into a smooth, taut ball by pulling dough edges underneath.
06 - Place dough balls in a greased 9x13-inch baking pan or on a lined baking sheet, leaving slight space between rolls. Cover loosely and let rise for 45 to 60 minutes until puffy and nearly doubled.
07 - Preheat oven to 350°F.
08 - Whisk together egg and milk until combined. Gently brush the tops of each roll with the egg wash, being careful not to deflate the dough.
09 - Bake for 18 to 22 minutes until rolls are deeply golden brown and sound hollow when tapped on the bottom.
10 - Let rolls cool in the pan for 5 to 10 minutes before transferring to a wire rack. Serve warm or at room temperature.