In the bowl of a stand mixer, stir together the warm milk, yeast, and ¼ teaspoon of the sugar. Let the mixture stand until it's very foamy, about 5 minutes.

Add the melted butter, eggs, and the remaining ¼ cup sugar. Attach the paddle to the mixer and mix on low speed until mostly combined. Add 2 cups of the flour and the salt. Continue mixing on low speed until the mixture is fully combined and smooth, about I minute.

Switch to the dough hook attachment and add the remaining 2¾ cups of four, ¼ cup at a time, kneading until a smooth dough ball forms and pulls away from the sides of the bowl, about 12 minutes. (The dough may still feel slightly sticky. Add a tablespoon of four as needed if the dough is too sticky and isn't pulling away from the bowl.) Add the cheese and rosemary and mix just until well combined, about 1 minute.

Coat a large bowl with the oil and transfer the dough to the bowl, turning to coat. Shape the dough into a ball, cover, and let rise in a warm spot until doubled in size, about 45 minutes.

Butter a 9x13-inch baking pan and set aside.

Turn the dough out onto a lightly floured surface. Cut into twelve equal pieces. Shape each piece into a ball and place in the prepared baking pan. Cover and let rise until doubled in size, about 30 to 45 minutes.

Preheat the oven to 350°F.

Brush the tops of the rolls with the beaten egg and bake for 25 minutes, until the tops are golden brown. Immediately brush the warm rolls with melted butter. Sprinkle with the flaky salt and rosemary. Let the rolls cool for 10 minutes before removing or serving.