Bring a large pot of water to a boil. Add 2 tablespoons of the salt and return to a rolling boil. Add the pasta and cook until al dente according to package directions.
While the pasta cooks, prepare the sauce: In a small bowl, whisk together the whole eggs, egg yolk, and Pecorino Romano and season with salt and pepper. Set aside.
Heat the olive oil in a 12-inch skillet over medium heat. Add the ham and cook until crisp, about 5 minutes.
Reduce the heat to low and add the pasta and 1/2 cup of pasta water, tossing together to coat. Remove the pan from the heat and add the egg mixture, tossing the pasta rapidly to ensure the egg doesn’t scramble.
Add the peas and toss again until glossy, adding 1/4 cup of pasta water or more (up to 1 cup), as needed to loosen up the sauce.
Plate in a large bowl and sprinkle with the parsley. Serve with additional cheese, if desired.