In a large bowl, whisk the eggs and egg yolks. Add the heavy cream, Parmesan, lemon zest and parsley, and whisk to blend. Set aside.
Cook the spaghetti in the salted boiling water until just al dente according to package directions. Drain.
Heat two teaspoons of olive oil in a heavy large frying pan over medium heat. Add the prosciutto, garlic and shallot, and saute, stirring occasionally, until fat begins to render, about 5 minutes. Remove from the pan and set aside.
Heat the remaining 1 tablespoon of olive oil in the same pan. Add the chicken, season with salt and pepper, and stir-fry on medium heat until the chicken is cooked through, about 7 minutes. Turn the heat on medium-low. Add prosciutto and stir to combine. Next, add the spaghetti, egg and cream mixture and peas, and toss over medium-low heat until the sauce coats the pasta thickly, about 4-5 minutes (sauce will thicken as it stands). Do not boil! Taste, and season with salt and pepper, as desired.
Divide the pasta between four bowls and finish each with a sprinkling of Parmesan cheese. Enjoy!