In a medium bowl, whisk eggs, Parmigiano, parsley, 1/2 teaspoon salt and 1/8 teaspoon pepper.
Heat two tablespoons of the oil in a 10″ ovenproof nonstick skillet over medium-high heat. Add the potato and 1/4 teaspoon salt and cook, stirring occasionally, until browned on all sides (6 to 7 minutes). Transfer to a bowl with a slotted spoon.
Reduce the heat to medium. Add the onion and cook, stirring frequently, until it softens and begins to brown. Stir in the asparagus, garlic, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Cook, stirring frequently, until the asparagus is bright green and crisp-tender. Lower the heat to low and add the egg mixture and the potatoes, stirring until the ingredients are combined. Add the cheddar and stir until well distributed. Cook without stirring until the eggs have almost set.
Meanwhile, position an oven rack 6″ from the broiler element and heat 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. Let rest for 5 minutes. Transfer to a cutting board, cut into wedges, and serve.
