In a small bowl combine paprika, smoked paprika, onion powder, garlic powder, dried oregano, dried thyme, cayenne pepper, black pepper and kosher salt. Stir to blend thoroughly.

Season shrimp

Place the shrimp in a medium bowl. Drizzle with 2 tablespoons olive oil and toss to coat. Sprinkle the spice rub over the shrimp and toss until each shrimp is evenly coated.

Heat pan

Heat a large heavy skillet (cast iron preferred) over medium-high heat until very hot, about 3–4 minutes. The pan should shimmer but not smoke heavily.

Cook shrimp

Add 1 tablespoon unsalted butter and swirl to coat the pan. Add the shrimp in a single layer without crowding (work in two batches if necessary). Cook 1–2 minutes on the first side until blackened and opaque at the edges, then flip and cook another 30–60 seconds until shrimp are fully opaque and firm. Total cook time per batch ~2–3 minutes. Avoid overcooking.

Finish

Transfer cooked shrimp to a warm plate. Squeeze fresh lemon wedges over the shrimp to taste. Keep warm while you cook remaining shrimp.

Cocktail Sauce

Combine sauce

In a small bowl combine ketchup, prepared horseradish, Worcestershire sauce, fresh lemon juice, hot sauce (if using), freshly ground black pepper and salt. Stir until smooth and taste; adjust horseradish for heat and salt or lemon for brightness.

Chill

Cover and chill the sauce for at least 10 minutes to allow flavors to meld. Serve cold or at room temperature alongside the blackened shrimp.