Fill a large pot with water, add a pinch of salt, and bring it to a rolling boil over high heat.
Carefully add the shrimp to the boiling water, stirring gently, and cook for 2-3 minutes until they turn a bright pink and are firm to the touch.
Using a slotted spoon, transfer the cooked shrimp immediately into a bowl of ice water to halt the cooking process and set their firm texture.
While the shrimp chill, combine ketchup, horseradish, lemon juice, Worcestershire sauce, and smoked paprika in a small saucepan.
Whisk the sauce ingredients together until smooth, then gently heat over low until just warmed through—about 1 minute—then remove from heat and set aside to cool slightly.
Drain the chilled shrimp well, pat dry with paper towels, and arrange them neatly on a serving platter.
Pour the warm cocktail sauce into a small bowl or serve it on the side for dipping.
Squeeze fresh lemon wedges over the shrimp and sauce for an extra burst of citrus flavor, and garnish with additional lemon if desired.
Chill the assembled shrimp cocktail in the refrigerator for at least 30 minutes before serving to keep the shrimp cold and refreshing.
Serve the shrimp chilled with the smoky spicy cocktail sauce on the side or drizzled over the shrimp for an impressive appetizer.
