Bring a pot of water to boil. When it's boiling, add salt. Throw the pasta in and cook until it's al dente. Be sure to reserve some pasta water.
While waiting for pasta water to boil, heat a large pan over medium heat. Add the olive oil and let it heat up.
When the olive oil is shimmering, add the chopped onion and let it cook until soft, 3-4 minutes. Add the prosciutto and let it cook down until slightly crispy, another 3-4 minutes.
Optional step: Deglaze with a splash of white wine.
When the alcohol from the white wine evaporates off (a few mins, if using), add the peas to the prosciutto/onion and cook for a few minutes. If pasta is not ready by this time, turn the heat on the pan down to low.
While peas are cooking, make the egg mixture: in a bowl, beat egg and egg yolks together, with parmigiano reggiano, freshly ground black pepper to taste, and a splash of pasta cooking water. Set aside.
When pasta is cooked to al dente, add it to the pan (still over medium heat), along with a ladle of pasta water.
Toss the pasta together with the prosciutto and peas for a few minutes, and remove from the heat.
Add the egg mixture to the pasta and toss once more until pasta is evenly coated.
Plate and top with additional parmigiano reggiano. Enjoy!