Fill a medium pot 2/3 full will water and salt it liberally. Bring to a boil over high heat. Add the peas and cook just until they turn bright green, about 30 seconds, then immediately drain in a colander and rinse under cold water to stop the cooking and preserve their color. Set aside.
Fill a large pot about 2/3 full with water and salt it liberally. Bring to a boil over high heat.
Meanwhile, heat a large heavy sauté pan over medium-high heat. Add 1 tablespoon of the butter and let it melt, coating the pan. Add the prosciutto and cook, stirring with a wooden spoon, just to warm and infuse the butter with its flavor, about 1 minute. Stir in the cream and bring to a simmer, then lower the heat and simmer gently until it redues and thickens enough to coat the back of the spoon, about 5 minutes.
When the water comes to a boil, add the pasta and cook until al dente, about 2 minutes for fresh or 7 minutes for dried.
Meanwhile, stir the peas and remaining tablespoon of butter into the sauce, letting the butter melt. Drain the pasta and add it to the sauté pan. Toss to coat with the sauce, then stir in ½ cup of the cheese to melt and thicken the sauce.
Divide the pasta among individual plates, finishing with a few grinds of black pepper, a sprinkling of cheese and arugula.