If you forgot to thaw the shrimp in the fridge overnight, thaw the shrimp under cold running water for 5–7 minutes. Pat very dry with paper towels, then toss in a bowl with the garlic, olive oil, and a generous pinch of salt. Let it marinate at room temperature while you bring a pot of salted water to a boil.
Cook the pappardelle until just shy of al dente, about 7–8 minutes. Reserve 1/2 cup of pasta water before draining.
While the pasta cooks, heat a large skillet over medium-high heat. Add the marinated shrimp in a single layer and sear for 1–2 minutes per side, until pink and just cooked through. Transfer to a plate.
Lower the heat to medium and add the butter to the same skillet. Once melted, stir in the lemon zest, lemon juice, and red pepper flakes, if using. Pour in the cream and bring to a gentle simmer.
Add the Parmesan and pasta water, stirring until the sauce is smooth and slightly thickened. Return the shrimp and drained pasta to the pan and gently toss to coat. Let everything warm through for another 1–2 minutes.
Remove from heat, stir in the fresh basil, and serve immediately with extra Parmesan and lemon wedges.