Bring a large pot of water to a boil. Add pasta and peas; cook until the pasta is tender and the peas are cooked through, 10 to 12 minutes.
Meanwhile, heat oil in a large nonstick skillet over medium heat. Cook prosciutto, stirring, until it begins to brown, about 5 minutes. Add garlic and cook, stirring, until fragrant, 30 seconds. Add mushrooms and cook, stirring, until they release their juices and most of the liquid has evaporated, about 4 minutes. Sprinkle flour over the mushrooms; stir to coat. Add wine and let simmer for 1 minute. Add broth, return to a simmer and cook, stirring, until slightly thickened, about 3 minutes. Remove from the heat. Stir in cream and pepper.
Drain the peas and pasta; return to the pan, add the mushroom sauce and toss to coat.