In a medium bowl combine 1 cup ketchup, 2 tablespoons prepared horseradish, 2 tablespoons fresh lemon juice, 1 teaspoon Worcestershire sauce, 1 teaspoon hot sauce, 1 teaspoon smoked paprika, 1 teaspoon kosher salt, and 1/2 teaspoon freshly ground black pepper. Whisk until smooth and taste; adjust horseradish (add up to 1 more tablespoon) or hot sauce to preference. Chill in refrigerator at least 30 minutes to let flavors meld.

Shrimp & Cooking Liquid

Prepare poaching liquid

In a large saucepan combine 6 cups water, 1 tablespoon coarse kosher salt, 1 bay leaf, 8 black peppercorns, the halved lemon (cut side down), and 2 sprigs fresh parsley if using. Bring to a gentle simmer over medium heat; do not boil vigorously.

Poach shrimp

Once the liquid is simmering, add 1 pound peeled and deveined large shrimp (tails on). Maintain a gentle simmer; cook until shrimp turn pink and opaque and curl into a loose C shape, about 2–3 minutes depending on size. Do not overcook.

Shock shrimp

Prepare an ice bath: fill a large bowl with 4 cups ice and enough cold water to chill. Using a slotted spoon, transfer shrimp from poaching liquid to the ice bath immediately to stop cooking. Let chill 2–3 minutes until completely cool.

Drain and chill

Drain shrimp well on a paper towel-lined tray or colander. Refrigerate shrimp uncovered for 10–15 minutes to firm and cool further if needed before serving. Reserve some poaching liquid for reheating if serving warm (optional).

Garnish & Serve

Arrange chilled shrimp around a small bowl or ramekin filled with the chilled cocktail sauce. Garnish with fresh lemon wedges and sprinkle 2 tablespoons freshly chopped parsley over the shrimp. Serve immediately with extra lemon wedges and remaining sauce on the side.