Heat oil in a large skillet over medium heat. Add pancetta and cook, turning, until golden brown, about 5 minutes. Transfer to paper towel-lined plate; set aside.
Add onion, garlic, and mushrooms to pan. Cook, stirring, until tender, about 5 minutes. Add tomatoes and cook until they begin to break down, about 2 minutes.
Add wine and reduce by half, scraping up any bits from bottom of pan with a wooden spoon. Pour in milk and return pancetta to pan. Gently simmer until sauce has thickened, about 15 minutes.
Meanwhile, cook pasta in well-salted boiling water according to package instructions. Reserve 1/2 cup pasta water. Drain pasta and toss with sauce. Add reserved pasta water to desired consistency. Season with salt and pepper. Serve with grated Parmesan.