Bring a large pot of salted water to a boil. Add the pasta and cook until very al dente, about 8 minutes, adding the peas in the last 2 minutes of cooking.
Meanwhile, whisk the eggs, egg white, evaporated milk and 1/2 teaspoon salt in a bowl.
Drain the pasta and peas; wipe the pot dry and return the pasta and peas to the pot. Add the butter and stir to coat. Add the cheddar, mozzarella, fontina and the egg mixture. Place the pot over medium-low heat and cook, stirring constantly, until thickened, 8 to 10 minutes, adding the ham halfway through cooking (don't let the mixture boil or the eggs will scramble). Remove the pot from the heat and stir in the parmesan.