Hold off on tossing in the prosciutto crisps and arugula until just before serving to keep them crisp.
Preheat oven to 400 degrees. Arrange prosciutto in a single layer on a parchment paper-lined rimmed baking sheet. Brush lightly with oil, and bake until crisp, about 10 minutes. Let cool completely.
In a large pot of boiling, salted water, cook tortellini according to package instructions. Add peas, then immediately drain. Let cool 10 minutes, then toss in a large bowl with lemon zest and juice, and olive oil. Season with salt and pepper.
Break prosciutto into large pieces and toss with pasta along with arugula. Serve in bowls topped with cheese.
To make this salad ahead, prepare the pasta salad through step 2, then cover and refrigerate for up to 1 day. Bring to room temperature before continuing with recipe.