Preheat oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish.
Cook linguine in salted boiling water according to package directions until al dente. Drain and set aside.
Melt butter in a large skillet over medium heat. Add minced garlic and red pepper flakes; cook for 1 minute until fragrant.
Add shrimp to the skillet and sauté for 3–4 minutes until pink and nearly cooked through. Remove shrimp with a slotted spoon and set aside.
Pour chicken broth into the skillet, scraping up browned bits from the bottom. Simmer for 2 minutes.
Stir in lemon juice, lemon zest, and heavy cream. Bring to a gentle simmer for 2 minutes.
Toss cooked pasta into the skillet, coating it thoroughly in the creamy sauce. Season with salt and black pepper.
Transfer the coated pasta to the prepared 9x13 baking dish. Arrange the seared shrimp on top.
Sprinkle Parmesan and mozzarella cheese evenly over the dish.
Bake for 15–20 minutes until the cheese is melted, bubbly, and lightly golden brown.
Garnish with fresh chopped parsley and serve warm.