Warm the oil in a large skillet over medium heat until it is shimmering but not smoking. Add chorizo and break into small pieces using a wooden spoon or potato masher. Brown the chorizo until it is fully cooked and starts to leach out some of its fat. Add the green onion and salt, reduce the heat to medium-low, and cook, stirring every few minutes. Take off heat and set aside.

Preheat the oven to 325 degrees. Beat the eggs and yolks together in a large bowl. Stir in the cream, the Gruyère, 3 tablespoons of the Parmigiano, potato slices, and chorizo green onion mixture, salt, and pepper to taste.

Heat 1 tablespoon oil in a 10-inch ovenproof nonstick skillet over medium-high heat. Add the egg mixture and, with a rubber spatula or wooden spoon, stir to scramble the frittata for 30 seconds, scraping the bottom of the pan and breaking up any large egg curds. The frittata will still be mostly wet, with evenly dispersed cooked egg throughout. Remove the pan from the heat, smooth the top of the frittata, and gently tap the pan on the counter to settle it.

Slide the pan into the oven and bake for 20 to 30 minutes, or until the frittata is firm to the touch. Remove from the oven and turn out onto a large platter (or cut it into wedges in the pan using a nonabrasive spatula). Garnish with the remaining 1 tablespoon Parmigiano and a drizzle of olive oil. Serve hot, warm, or at room temperature with salad and some toast and boom you have a perfect week day meal or weekend brunch.