Pour 1 tsp. extra-virgin olive oil into a cold large skillet; arrange 2 thin slices prosciutto in an even layer in pan. Set over medium-low heat and cook, turning occasionally, until prosciutto is crisp, 5–7 minutes. Transfer to paper towels to drain; set aside. Reserve skillet.
Cook one 8-oz. package Trader Joe’s egg pappardelle pasta or 8 oz. other dried pappardelle in a large pot of boiling salted water until al dente. Drain pasta, reserving 1 cup pasta cooking liquid.
Meanwhile, heat remaining 1 Tbsp. extra-virgin olive oil in reserved skillet over medium-low. Cook 2 garlic cloves, finely chopped, stirring, until fragrant, about 20 seconds. Pour in ½ cup heavy cream, increase heat to medium, and bring to a simmer, scraping up any browned bits stuck to bottom of pan with a wooden spoon. Cook, stirring often, until sauce is thickened, about 2 minutes; reduce heat to low. Using a Microplane, finely grate zest of 1 lemon into sauce. Cut lemon in half and squeeze in juice from 1 half. Cut remaining lemon half into wedges; set aside for serving. Stir in 2 Tbsp. finely grated Parmesan.
Add pasta and ¼ cup pasta cooking liquid to sauce. Cook, tossing often and adding more pasta cooking liquid if needed (you’ll probably use about ¾ cup), until pasta is coated and glossy. Season with freshly ground black pepper; taste and season with kosher salt if needed.
Divide pasta among bowls. Tear reserved prosciutto over and season with more pepper. Serve with reserved lemon wedges.