Bring a large pot of generously salted water to a boil. Cook linguine according to package directions for al dente. Reserve 1 cup pasta cooking water before draining.

Pat shrimp completely dry with paper towels on both sides. Season lightly with salt and black pepper.

Heat 1 tablespoon butter and olive oil in a large skillet over medium-high heat until the butter foam subsides. Add shrimp in a single layer. Cook 90 seconds undisturbed, flip, cook 60 seconds more. Remove shrimp to a plate.

Reduce heat to medium. Add remaining 2 tablespoons butter to the same pan. Once melted, add garlic and chili flakes. Stir constantly for 60 to 90 seconds until fragrant and pale gold.

Pour in chicken broth, scraping any browned bits from the pan base. Raise heat to bring to a simmer and reduce for 2 to 3 minutes until the liquid is reduced by roughly a third.

Turn off heat entirely. Add the cold finishing butter, lemon juice, and lemon zest. Stir briskly in circles for 30 seconds until the sauce turns opaque and glossy.

Return shrimp to the pan with all resting juices. Add drained linguine. Toss with tongs for 30 to 60 seconds, adding pasta water 2 tablespoons at a time if needed.

Divide among warm bowls. Scatter parsley over each portion. Serve immediately with extra lemon wedges and Parmesan if using.