Heat a large skillet over medium-high heat. Add the oil and heat. Add the chopped prosciutto and cook, stirring constantly, until crispy and deep golden brown, about 4 minutes. Using a slotted spoon, remove the prosciutto to a paper towel-lined plate. Reduce the heat to low. Add the garlic to the oil and cook, stirring often, until lightly golden, about 3 minutes. Add the peas, pepper flakes and 1/8 teaspoon salt and turn off the heat.
Add the pasta to the boiling salted water and cook for 2 minutes less than package directions, about 9 minutes.
Return the skillet with the garlic oil to medium heat. Using tongs, remove the pasta from the water and add directly to the skillet with the oil. Sprinkle the bare pasta with 1 cup Parmigiano-Reggiano. Add 1/2 cup pasta water and toss well to coat the pasta and form a creamy sauce. Add the remaining 1 cup Parmigiano-Reggiano and up to 1/2 cup more pasta water to maintain a creamy sauce. Serve sprinkled with the crispy prosciutto, more Parmigiano-Reggiano and another drizzle of olive oil, if desired.