Stir together yeast, honey, and ⅓ cup warm milk in mixer bowl and let stand until foamy, about 5 minutes. (If mixture doesn’t foam, start over with new yeast.) Whisk together flour (2 ½ cups), cheese, and salt, then mix into yeast mixture along with remaining ⅓ 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.)