Using a stand mixer fitted with paddle attachment, stir together yeast, honey, and 1/3 cup warm milk in mixer bowl and let stand until foamy, about 5 minutes. (If mixture doesn't foam, start over with new yeast.)
In a separate bowl, whisk together 2-1/2 cups flour, cheese, and salt. Mix into yeast mixture along with remaining 1/3 cup warm milk at low speed. Increase speed to medium and beat in 2 eggs, 1 at a time, beating well after each addition, then beat, scraping down side of bowl occasionally, until a very soft dough forms, about 3 minutes. Beat in butter, 1 tablespoon at a time, until dough is elastic, about 2 minutes. (Dough will be very sticky.) Scrape dough into center of bowl and sprinkle with remaining 2 tablespoons flour. Cover bowl with plastic wrap and a kitchen towel and let dough rise in a draft-free place at warm room temperature until doubled, 1-1/2 to 2 hours.
Punch down dough (do not knead) and turn out onto a floured surface. Cut dough into 12 equal pieces and roll each into a ball by cupping your hand and pushing dough against work surface as you roll in a circular motion. Arrange rolls 1 inch apart in a buttered 9x13x2-inch pan and cover with a kitchen towel (not terry cloth). Let dough rise in a draft-free place at warm room temperature until doubled and dough fills pan, 1 to 1-1/2 hours.
Place the oven rack in the middle. Preheat oven to 375 degrees. Whisk together remaining egg with water and brush on tops of rolls. (You will have leftover egg wash.) Bake until golden brown, 20 to 25 minutes. Loosen edges of rolls from pan with a sharp knife and invert rolls onto a rack, then quickly re-invert and cool at least 20 minutes.