Activate the Yeast: In a medium mixing bowl, combine the warm milk, active dry yeast, and sugar. Stir gently and let the mixture sit undisturbed for about 5–7 minutes, or until a light foam forms on the surface. This is a sign the yeast is alive and ready to rise.
Mix and Form the Dough: In a large mixing bowl (or the bowl of a stand mixer), add the bread flour, salt, softened butter, and egg. Pour in the activated yeast mixture. Use a spatula or your hands to mix everything together until a shaggy, sticky dough forms and starts pulling away from the sides of the bowl.
Knead Until Smooth: Turn the dough out onto a lightly floured surface. Knead by hand for about 8–10 minutes, or until the dough becomes smooth, elastic, and slightly tacky but not sticky. If using a stand mixer, knead on medium speed for about 6–8 minutes with the dough hook.
Let the Dough Rise: Shape the dough into a ball and place it into a lightly oiled bowl. Cover with plastic wrap or a clean towel, and let it rise in a warm, draft-free place for 1 to 1½ hours, or until doubled in size. The dough should feel soft and airy.
Divide and Shape Rolls: Once the dough has risen, gently deflate it by pressing down with your hands. Transfer it to a clean surface and divide it into 9 equal portions. Shape each piece into a smooth ball by pulling the sides down and tucking underneath, pinching the seam at the bottom.
Arrange for Second Rise: Place the dough balls evenly into a greased 9x9-inch square baking pan, leaving some space between each one. Cover loosely with plastic wrap and allow to rise again for about 30–40 minutes, or until noticeably puffy and nearly touching.
Preheat and Prepare: Preheat your oven to 350°F (175°C). In a small bowl, beat 1 egg to make an egg wash. Gently brush the tops of the rolls with the egg wash to give them a shiny, golden finish after baking.
Bake to Golden Perfection: Bake the rolls in the preheated oven for 25–30 minutes, or until the tops are beautifully golden and sound hollow when tapped. Keep an eye on them during the last few minutes to avoid over-browning.
Butter and Finish: Remove the rolls from the oven and immediately brush the tops with melted butter for an extra-rich flavor. Optionally, sprinkle a pinch of flaky salt over the tops while still warm. Serve fresh and enjoy!