In a large pot, combine 6 quarts of water with 2 tablespoons kosher salt and bring it to a boil.
Grate the Parmesan and mozzarella cheese. Carefully separate four egg yolks and set aside.
Once boiling, add the pasta and cook until the pasta is just about al dente, about 7 minutes; then add peas and spinach and cook for 1 minute. Reserve 1 cup cooking water, and then drain the pasta and vegetables.
In a skillet, melt the butter, then stir in the cheeses, ¼ cup pasta water, and ¼ teaspoon kosher salt. Stir in the pasta and vegetables until creamy over low heat, adding more pasta water if necessary (note that the mozzarella will stick together in some places).
To serve, top each pasta serving with a whole egg yolk and additional Parmesan cheese, and stir the yolk into the pasta at the table. (If you are uncomfortable serving egg yolks at the table, stir the egg yolks into the pasta in the skillet to heat them through.) Serve immediately.
Note: The mozzarella cheese can become gummy the longer the pasta sits, so eat immediately if possible. Leftovers can be reheated in a skillet, but may not have the same creamy texture.