Put a pot of water on to boil. Salt it well and boil the pasta following the package directions.

Meanwhile, trim, wash, and thoroughly dry the arugula; chop it coarsely and set aside. In a large saucepan over medium heat, melt 4 tablespoons of the butter. Add the garlic and saute until barely golden, about 2 minutes. Add the peas and saute until tender-crisp, another 3 to 4 minutes if using fresh or 1 minute if using thawed frozen. Season lightly with salt and pepper. With a slotted spoon, remove the peas and garlic from the pan and set aside. Leave as much liquid in the pan as possible and add the olive oil to the pan. Add the prosciutto and cook until very lightly browned, about 5 minutes. Add the arugula in batches, stirring and adding handfuls as it wilts. Add the lemon juice. Cook for only a few minutes, just until all the greens have wilted.

When the pasta is done, reserve 1 cup of the cooking water and drain the pasta. In a large bowl, toss the pasta with the peas, prosciutto, arugula, and grated Parmigiano, along with 1/2 cup of the reserved cooking water. Season with more ground pepper; toss. Taste and adjust the seasonings if needed. If the pasta seems dry, toss with more of the reserved cooking water. To serve, garnish each portion with the parsley and the Parmigiano shavings.