In a mixer bowl, combine yeast, sugar, and 1/3 cup of warm milk. Let stand until foamy, about 5 minutes.

In a separate bowl, whisk together flour, grated Parmesan cheese, and salt.

Add the flour mixture to the yeast mixture, along with the remaining 1/3 cup of milk. Mix at low speed.

Increase speed to medium and beat in two eggs, scraping down the sides of the bowl occasionally, until a soft dough forms (about 3 minutes).

Beat in softened butter until the dough is elastic (approx. 2 minutes). The dough will be very sticky.

Scrape down the bowl, cover with plastic wrap and a kitchen towel, and let rise in a warm room until doubled (1 1/2 - 2 hours).

Punch down the dough (do not knead) and turn it out onto a floured surface.

Cut the dough into 12 equal pieces and roll each into a ball.

Arrange the balls 1 inch apart in a buttered 9-by-2 inch round cake pan. Cover with a kitchen towel.

Let the dough rise in a draft-free place at a warm room temperature until doubled and the dough fills the pan, 1-1/2 hrs.

Preheat oven to 375°F (190°C) with rack in middle.

Whisk together remaining egg with water and brush on tops of rolls (you will have leftover egg wash).

Bake until deep golden brown, 20 to 25 minutes.

Loosen edges of rolls pan when the come out of the oven, then let cool for several minutes before serving.