Activate the Yeast: Combine the warm milk, warm water, granulated sugar, and active dry yeast in a small bowl. Let the mixture sit for about 5–10 minutes, until the yeast becomes foamy on top.
Make the Dough: Add the flour and salt to a large mixing bowl. Pour in the yeast mixture, melted butter, and egg. Stir until a soft, shaggy dough begins to form.
Knead the Dough: Transfer the dough to a lightly floured surface and knead for about 7–10 minutes, until smooth, soft, and elastic. The dough should feel slightly tacky but not excessively sticky.
First Rise: Place the dough in a lightly greased bowl and cover it with a clean kitchen towel. Let it rise in a warm, draft-free area for about 1 hour, or until doubled in size.
Divide the Dough: Gently press down the risen dough to release excess air. Transfer it to a lightly floured surface and divide it into 8 equal portions.
Shape the Rolls: Shape each portion into a smooth, slightly elongated roll. Place the shaped rolls on a parchment-lined baking sheet, leaving enough space between them for expansion.
Second Rise: Cover the shaped rolls loosely with a clean kitchen towel and let them rise for about 25–30 minutes, or until noticeably puffy.
Apply the Egg Wash: Preheat the oven to 375°F. Whisk the egg with the water, then gently brush the mixture over each risen roll. Sprinkle with sesame seeds or poppy seeds if desired.
Bake Until Golden: Bake for about 16–18 minutes, or until the rolls are puffed and beautifully golden brown on top.
Cool and Serve: Transfer the rolls to a wire rack and let them cool for several minutes. For an especially soft crust, lightly brush the warm rolls with melted butter before serving.