Beat together eggs, Parmesan, chives, cayenne, and salt and pepper to taste.
Heat 2 Tbs. oil in a large non-stick, oven-proof skillet over medium-high heat. Saute potatoes until lightly browned, about 8 minutes. Remove potatoes from pan with a slotted spoon
Lower heat and add additional oil if necessary. Add onion and mushrooms and cook, stirring, until onion begins to brown, about 8 minutes. Add asparagus and garlic and cook until asparagus is just barely tender, about 3 minutes. Stir in prosciutto and cook 1 additional minute.
Reduce heat to low and stir in eggs and potatoes. Stir in Gruyere. Cook until eggs are nearly set in the center, about 15 minutes.
Preheat broiler and place skillet under element. Broil until eggs are set and lightly golden. Remove from oven and let rest 5 minutes. Garnish with additional chives.