In a medium bowl, whisk the eggs, Parmigiano, parsley, 1/2 tsp. salt, 1/8 tsp. pepper, and the cayenne.
Heat 2 Tbs. of the oil in a 10-inch ovenproof nonstick skillet over medium-high heat. Add the potato and 1/4 tsp. salt and cook, stirring occasionally, until browned on several sides, 6 to 7 minutes. Transfer to a bowl with a slotted spoon. Reduce the heat to medium. If the pan is dry, add the remaining 1 Tbs. oil. Add the onion and cook, stirring frequently, until it softens and begins to brown, 4 to 5 minutes. Stir in the asparagus, garlic, 1/4 tsp. salt and 1/8 tsp. pepper. Cook, stirring frequently, until the asparagus is bright green and crisp-tender, 3 to 4 minutes. Lower the heat to medium low and add the egg mixture and the potatoes, stirring until the ingredients are combined, 10 to 15 seconds. Add the Cheddar and stir until well distributed. Cook without stirring until the eggs have almost set, 10 to 12 minutes. (The center may still be loose but should be bubbling a little; the sides should be set.) Meanwhile, position an oven rack 6 inches from the broiler element and heat the broiler to high.
Transfer the skillet to the oven and broil until the eggs have set completely and the top of the frittata is golden brown, 1 to 3 minutes. Let rest for 5 minutes. Transfer to a cutting board, cut into four wedges, and serve.
This frittata is a great make-ahead dinner. Let it cool to room temperature and refrigerate until ready to serve. It can be warmed in the microwave or served at room temperature.