Heat the oven to 375°F (190°C) to get it ready for baking.
Cook the linguine or spaghetti following the package directions until just tender, then drain and set aside.
In a skillet over medium heat, melt butter and then add minced garlic with red pepper flakes. Cook briefly until the aroma is released, about 1 minute.
Add shrimp to the pan and cook until they turn pink and opaque, around 3 to 4 minutes, then remove them from the skillet and keep aside.
Pour the white wine or chicken broth into the pan, scraping up any browned bits, and let it bubble gently for 2 minutes.
Mix in lemon juice, lemon zest, and heavy cream; warm gently until the sauce begins to simmer.
Return the pasta to the skillet and stir well to coat it completely with the creamy sauce; season with salt and pepper to your liking.
Spread the sauced pasta evenly in a baking dish, and lay the cooked shrimp on top.
Sprinkle the grated Parmesan and shredded mozzarella over everything to create a golden crust.
Bake in the oven for 15 to 20 minutes until the cheese has melted and the surface is bubbly and lightly browned.
Take the dish out of the oven, scatter chopped fresh parsley over the top, and serve while still warm for the best flavor.