Cook the pasta. Bring a large pot of water to a boil, then add a generous 1 Tbsp 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.
Crisp the pancetta. 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, about 3-4 minutes. Use a slotted spoon to transfer pancetta to a paper towel lined plate.
Sauté the mushrooms. If the pan looks dry, add an additional 2 Tbsp olive oil. Add 8oz sliced shiitake mushrooms to the pan and cook undisturbed until golden brown on the first side, about 3-4 minutes. Stir and cook an additional 3-4 minutes. Add chopped shallot and 1 cup of English peas, then season with 1 tsp Kosher salt and ¼ tsp freshly ground black pepper. Cook for 3 minutes, or until the peas are cooked through, but have a slight crunch to them. Add 2 cloves chopped garlic, then cook about 1 minute more.
Make the sauce. Pour in ½ cup white wine and 1 Tbsp white wine vinegar, then use a spatula to scrape up any browned bits from the bottom of the pan. Allow to cook down for 1 minute, then return the cooked pasta to the pan along with ⅓ cup heavy cream, ½ cup reserved pasta water, chopped mint leaves, ¼ cup grated Parmesan, and 2 Tbsp butter. Stir until the sauce is smooth and creamy, then taste and adjust seasoning as needed.
Serve immediately! Serve with additional fresh mint, Parmesan cheese, and a pinch of flaky sea salt.