Step 1: Prepare the Garlic Butter Mixture
Preheat the oven to 325°F (163°C).
Line a sheet tray with a silicone mat or parchment paper.
Arrange 6 shallow 5-inch round ramekins on the sheet tray.
In a medium mixing bowl, whip the softened butter with an electric mixer until creamy.
Add the roasted garlic paste, minced onion, prosciutto, lemon juice, cognac, tarragon, salt, and pepper to the bowl. Mix on low speed.
Slowly pour in the olive oil to emulsify the mixture, creating a creamy consistency similar to mayonnaise.
Switch to a spatula and gently fold in the seasoned breadcrumbs until well combined.
Step 2: Assemble the Gratin
Pour 1 tablespoon of white wine into the bottom of each ramekin.
Evenly distribute the shrimp into the ramekins on top of the wine.
Spoon the garlic butter mixture evenly over the shrimp, spreading it out into an even layer.
Step 3: Bake and Serve
Bake the tray in the preheated oven for 12-14 minutes, until the gratin is bubbly and the topping is slightly crisp.
Once out of the oven, squeeze a little extra lemon juice over the top of each serving.
Sprinkle additional chopped tarragon on top for garnish.