Add the sea salt to a large pot of water and bring to a boil. Cook the spaghetti according to package instructions until al dente. Use a heat-proof measuring cup to reserve 1 cup of pasta water before draining the pasta.
While the pasta cooks, add the olive oil to a deep-sided skillet over medium-high heat. Add the prosciutto and cook until crispy. Transfer the prosciutto to a paper towel-lined plate to drain.
Pour off the grease from the skillet into a bowl. Add 3 tablespoons of grease back to the skillet. If you don't have a full 3 tablespoons, add enough olive oil to reach the full amount. Place the skillet over medium heat.
In a medium bowl, whisk together the eggs and Parmigiano Reggiano cheese. Set aside.
Once the oil is warm, add the cooked pasta to the skillet. Then slowly pour the egg mixture in while stirring and turning the pasta to coat it. If it becomes a little dry (clumpy), add a couple of splashes of the reserved pasta water as needed to loosen up the sauce.
Add the cooked prosciutto and peas, and toss together until the peas are warmed through. Season with salt and pepper to taste.
Serve warm with a little extra Parmigiano Reggiano sprinkled over the top.
Notes
If you can't find diced prosciutto, diced pancetta may be substituted. You can also use Parmesan instead of Parmigiano Reggiano if desired.