Combine water, yeast, and 1 teaspoon sugar in the bowl of a standing electric mixer. Stir to dissolve and set aside until bubbly, 5 to 10 minutes.
Add milk, butter, salt, remaining sugar, egg, and 2 cups of the flour. Mix until well combined, and then add remaining flour, 1/2 cup at a time, until dough just starts to clear the sides of the bowl. Switch to the dough hook and knead until a soft, elastic dough forms. Transfer to a lightly oiled bowl, cover, and let rise in a warm place for about 20 minutes.
Parchment-line two baking sheets. Turn the dough out and divide into 24 equal pieces. Shape pieces into rounds. Tuck a cube of cheese into each round and pinch the dough tightly together at the bottom, working to shape each one into a tight ball. Place rolls, pinched-side down on prepared baking sheets, spacing them about 1 inch apart. Cover with plastic wrap and let rise again in a warm place for about 20 minutes.
Preheat oven to 375° F. Just before baking, brush the tops of the rolls with melted butter. Bake for 15 to 18 minutes, switching baking sheets halfway through baking time, until the rolls are golden brown. Remove from the oven and cool slightly before serving.