1. In a 3- to 4-quart pan, combine 1 cup water, butter, and pepper; bring to a boil over high heat. Add flour all at once and stir quickly until mixture pulls away from pan sides and masses together. Remove from heat and stir until flour is incorporated and mixture is smooth. Let cool about 5 minutes, stirring occasionally.
2. To mix by hand, add eggs, one at a time, to warm butter-flour mixture in pan, beating with a wooden spoon after each addition until dough is smooth and satiny. Stir in shredded asiago cheese.
To mix with an electric mixer or food processor, scrape warm butter-flour mixture into a bowl. Add eggs, one at a time, beating or whirling after each addition just until smooth; stir in shredded asiago cheese, do not overmix.
3. On a buttered 12- by 15-inch baking sheet, drop eight equal portions (about 1/3 cup each) of asiago-pepper dough, side by side with edges touching, in an 8-inch circle. Sprinkle with another 1/4 cup shredded asiago.
4. Bake in a 375° regular or convection oven until browned, 40 to 50 minutes. Pierce each puff in several places with a wooden skewer and return to oven. Bake until well browned and crisp, 5 to 7 minutes longer. With a spatula, slide ring off pan. Serve hot, breaking off puffs to eat.