Bring 4 cups of salted water to a rolling boil in a large pot. Add linguine and cook until al dente, about 9–11 minutes. Reserve 1/2 cup pasta cooking water, then drain the pasta.

While pasta cooks, pat shrimp very dry and season with salt and pepper. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add shrimp in a single layer and sear for 1–2 minutes per side until pink and just cooked through. Transfer shrimp to a plate and set aside.

Reduce heat to medium, add remaining tablespoon of olive oil and 2 tablespoons of butter to the skillet. Add minced garlic and shallot and cook for about 1–2 minutes until fragrant and softened but not browned. If using red pepper flakes, add them here.

Pour in white wine (or broth) to deglaze the pan, scraping up brown bits. Let simmer for 2–3 minutes until reduced by about half.

Stir in heavy cream and bring to a gentle simmer. Let the sauce thicken for 3–5 minutes, stirring occasionally.

Reduce heat to low and whisk in grated Parmesan until melted and smooth. Stir in lemon juice, and adjust salt and pepper to taste.

Add drained pasta and seared shrimp to the skillet, tossing to coat. If needed, add up to 1/4 cup reserved pasta water to loosen the sauce. Heat briefly until everything is warmed through, about 1 minute.

Remove from heat, sprinkle with chopped parsley and extra Parmesan. Serve immediately while the sauce is silky and warm.