Bring a large pot of water to a boil, then add a generous Kosher salt. Add pasta and cook according to package directions, or until al dente. Reserve about 1 cup of pasta water before draining the pasta.
While the pasta is cooking, heat 1 Tbsp olive oil in a large skillet over medium-high heat. When hot, add 8oz cubed pancetta and cook, stirring often, until mostly crispy. Use a slotted spoon to transfer pancetta to a paper towel lined plate.
If the pan looks dry, add an additional olive oil. Add 8oz sliced shiitake mushrooms to the pan and cook undisturbed until golden brown on the first side. Stir and cook an additional 3-4 minutes.
Add chopped shallot and 1 cup of English peas, then season with Kosher salt and freshly ground black pepper. Cook until the peas are cooked through, but have a slight crunch to them. Add chopped garlic, then cook about 1 minute more.
Pour in white wine and white wine vinegar, then use a spatula to scrape up any browned bits from the bottom of the pan. Return the cooked pasta to the pan along with heavy cream, reserved pasta water, chopped mint leaves, grated Parmesan, and butter. Stir until the sauce is smooth.
Serve with additional fresh mint, Parmesan cheese, and a pinch of flaky sea salt.