Preheat the oven to 425°F (220°C) . Lightly grease a shallow 2-quart baking dish with 1 tablespoon (14 g) of the butter. Pat the shrimp dry with paper towels. Season them with ½ teaspoon of the kosher salt and all of the black pepper, then arrange them in a single layer in the prepared dish.

Melt the remaining 2 tablespoons (28 g) butter in a large skillet over medium heat. Add the minced garlic and cook for about 30–45 seconds, stirring constantly. The garlic should smell fragrant but should not turn dark brown.

Sprinkle the flour over the garlic butter: Whisk continuously for about 1 minute. The mixture should look smooth and lightly foamy, without visible dry flour.

Slowly whisk in the heavy cream, followed by the seafood stock or chicken broth. Add the remaining ¼ teaspoon kosher salt. Cook over medium heat for 3–4 minutes, whisking frequently, until the sauce lightly coats the back of a spoon. It should be creamy but still pourable.

Remove the skillet from the heat: Stir in the lemon zest, lemon juice, and 1 tablespoon (4 g) of the chopped parsley. Taste the sauce. It should be creamy, garlicky, and lightly bright from the lemon.

Pour the warm garlic cream sauce evenly over the raw shrimp. Use a spoon to make sure the sauce reaches between the shrimp rather than sitting only on top. The shrimp should remain mostly in a single layer.

In a small bowl, combine the Parmesan, Gruyère, panko breadcrumbs, melted butter, and smoked paprika. Mix until the breadcrumbs are lightly coated and the cheeses are evenly distributed.

Sprinkle the breadcrumb and cheese mixture evenly over the shrimp and sauce. Do not press it down firmly; a loose layer browns more evenly and stays crisper.

Bake at 425°F (220°C) for 10–12 minutes. The sauce should bubble around the edges, the topping should turn golden, and the shrimp should become opaque and curl into loose C-shapes. Check the thickest shrimp with an instant-read thermometer. It should reach 145°F (63°C) . If the topping needs more color but the shrimp are already cooked, broil for no more than 30–60 seconds while watching closely.

Remove the gratin from the oven and let it rest for 5 minutes. The sauce will thicken slightly as it stands.

Sprinkle the remaining 1 tablespoon (4 g) chopped parsley over the top. Serve warm, spooning some of the garlic cream sauce and crisp topping over each portion.