Chef John's New Orleans-Style Barbequed Shrimp

Peel shrimp and place into a mixing bowl; set shrimp shells aside in a saucepan.

Drizzle vegetable oil over shrimp and season with black pepper, smoked paprika, cayenne pepper, and seafood seasoning. Mix shrimp to coat with spices and cover bowl with plastic wrap; refrigerate shrimp to absorb flavors, at least 1 hour.

Place reserved shrimp shells in saucepan over medium-high heat with 1 tablespoon butter; cook and stir until shells are pink and fragrant, 1 to 2 minutes. Pour in chicken stock, bring to a boil, and reduce heat to low; simmer until shrimp shells have given off their flavor, 20 to 30 minutes.

Strain shrimp stock through a fine mesh strainer into a bowl and add lemon juice, Worcestershire sauce, and hot sauce. Stir to combine.

Place a large skillet over high heat until pan is very hot; sear shrimp in the very hot, dry pan until shrimp are browned, about 1 minute per side.

Stir 3 tablespoons cold butter, garlic, and minced rosemary into shrimp; cook and stir until shrimp are opaque in the middle and garlic is fragrant, 1 minute. Pour in shrimp stock.

Transfer shrimp from skillet to a bowl, using a slotted spoon; reserve sauce in skillet. Bring sauce to a boil and cook until reduced slightly, about 5 minutes. Adjust seasoning to taste. Return shrimp to pan, reduce heat to low, and warm through, about 1 minute. Serve shrimp drizzled with pan sauce; garnish with a rosemary sprig.