For the sauce
2 cups loosely packed fresh flat-leaf parsley leaves
¼ cup coarsely chopped onion
¼ cup extra-virgin olive oil
¼ cup red wine vinegar
4 cloves garlic, minced
1 teaspoon fresh oregano leaves
¼ teaspoon cayenne pepper, or to taste
1/8 teaspoon salt, or to taste
1/8 teaspoon freshly ground pepper, or to taste
For the frittata
1 tablespoon unsalted butter
1 tablespoon olive oil
1 large sweet or yellow onion, halved vertically, each half cut into 1/8-inch-wide lengthwise strips (about 2 cups)
½ cup freshly grated Parmesan cheese
Dash of salt
Dash of freshly ground pepper
Preheat the broiler.
To make the sauce, process all the ingredients in a food processor until smooth. Taste and adjust the seasoning. Set aside.
To make the frittata, preheat the oven broiler. Heat the butter and oil in a large ovenproof skillet, preferably nonstick, over medium-high heat. When the butter is melted, add the onion and cook, stirring occasionally, for about 5 minutes, or until the onion is translucent and the water has cooked off. Reduce the heat to low; continue to cook, stirring occasionally, until the onion strips are tender and nicely browned, about 20 minutes.
While the onion is cooking, lightly beat the eggs in a medium bowl; stir in the cheese, salt, and pepper. When the onions are cooked, scrape up the browned bits in the bottom of the pan and pour the egg mixture into the skillet. Cook, undisturbed, until the edges and bottom of the frittata are firm, about 10 minutes.
Transfer the skillet to a broiler rack about 4 or 5 inches from the heating element. Broil for about 3 minutes, or just until the top of the frittata is set and lightly browned; watch closely so the eggs do not overcook.
To serve, slice the frittata into 12 thin wedges. Serve warm or at room temperature; top each serving with about 2 teaspoons of the sauce.
The sauce will keep for up to 2 days in a covered container in the refrigerator. Bring to room temperature to serve with a freshly made frittata.