Bring a large pot of water to a boil. Add the spaghetti noodles and cook according to the package directions for al dente pasta. Drain.
Meanwhile, heat the olive oil in a large pan or casserole dish over medium-high heat.
Add the prosciutto and sauté, stirring occasionally, until the prosciutto has rendered its fat and has become crispy. This should take about 5 minutes. Remove and place on a paper towel-lined plate.
Add the peas to the hot pan and sauté for 1 to 2 minutes, just until they are heated through and have a touch of color. Remove to the plate with the prosciutto.
Turn the heat down to medium. Add the butter, minced garlic, heavy cream, milk, salt, and pepper to the pan. Simmer for 2 minutes, stirring often.
Turn the heat to low. Add the grated parmesan and stir until it is melted and well combined into the sauce.
Add the peas and prosciutto to the sauce. (See notes)
Toss the cooked spaghetti into the sauce and serve immediately.
Notes
For presentation, I like to keep some of the peas and prosciutto out of the sauce to sprinkle over the top before serving.