Melt butter in heavy large skillet over medium heat. Add carrots and sauté 3 minutes. Add stock; cook until carrots are tender, about 5 minutes. Using slotted spoon, transfer carrots to plate. Add frozen peas, 1 cup prosciutto and cream to skillet and simmer until liquid thickens slightly, about 3 minutes. Add 1/2 cup Parmesan, Fontina, mozzarella and Romano. Stir until cheeses melt.
Meanwhile, cook pasta in large pot of boiling salted water until tender but still firm to bite. Drain well. Add pasta to skillet and toss to coat. Season to taste with salt and pepper.
Transfer pasta to large bowl. Garnish with carrots. Sprinkle remaining 1/4 cup prosciutto, 1/4 cup Parmesan and parsley over and serve.