Cook spaghetti in a large pot of lightly salted boiling water until just tender, 8 to 10 minutes.
Meanwhile, heat oil in a large skillet over medium-low heat. Add garlic and crushed red pepper; cook, stirring, until fragrant but not colored, 1 to 2 minutes. Add prosciutto and cook, stirring often, until lightly browned, 2 to 3 minutes. Stir in roasted peppers and arugula; increase heat to medium-high. Cook, stirring often, until the arugula is wilted, 3 to 4 minutes.
Reserving 1/3 cup of the cooking water, drain the spaghetti and place in a warmed large bowl. Add the reserved water, arugula mixture, 1/3 cup Parmesan and pepper; toss to coat well. Sprinkle with walnuts and the remaining Parmesan and serve.
Ingredient note: We tasted several brands of whole-wheat spaghetti and preferred the appealing springiness and nutty flavor of Bionaturae, found in the natural-foods section of large markets and natural-foods stores.