Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain through a fine-mesh strainer and return to the pot.
While the pasta is working, in a medium skillet, melt 2 tablespoons butter over medium heat. Add the carrots and sugar; season with salt. Partially cover the pan; cook for 5 minutes. Stir in the scallion whites, season with pepper and cook for 3 minutes. Stir in the scallion greens and peas and cook for 1 minute. Add the vegetable mixture and cheese to the pasta and toss.
On a work surface, pound the pork slices thin; season both sides with salt and pepper. Top each pork slice with a sage leaf and wrap a prosciutto piece around each. In a large skillet, heat the EVOO and the garlic over medium heat. Working in 2 batches, add the pork and cook until brown, 2 minutes on each side, transferring the cooked meat to a platter.
Add the remaining 2 tablespoons butter to the skillet; cook until brown, 2 minutes. Stir in the parsley and lemon juice, then pour the pan sauce over the pork. Serve with the pasta.