Bring a large pot of salted water to a boil. Cook linguine according to package instructions until al dente (usually 8–11 minutes). Reserve 1/2–1 cup pasta water, then drain and set aside.

While pasta cooks, pat shrimp dry and season lightly with salt, pepper, and a pinch of paprika.

Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add shrimp and cook 2–3 minutes per side until pink and opaque. Remove shrimp to a plate (they’ll finish cooking in the sauce).

In the same skillet, reduce heat to medium and melt 2 tablespoons butter. Add minced garlic and shallot; sauté 30–60 seconds until fragrant (don’t burn).

Pour in heavy cream and milk, stirring to combine. Bring to a gentle simmer and let it reduce for 3–5 minutes until slightly thickened.

Gradually stir in shredded cheddar, a handful at a time, until fully melted and smooth. If sauce is too thick, loosen with reserved pasta water (start with 1/4 cup). If too thin, simmer 1–2 minutes or add the cornstarch slurry to thicken.

Stir in lemon zest, lemon juice, smoked paprika, and red pepper flakes (if using). Taste and adjust seasoning with salt and pepper.

Add cooked linguine and shrimp back into the skillet. Toss gently to coat everything in the sauce; heat through 1–2 minutes. If the sauce tightens up, add a splash more pasta water to achieve desired consistency.

Serve immediately, garnished with chopped parsley or chives and an extra sprinkle of cheddar or lemon zest if desired.