In a medium bowl, add 1 cup ketchup, 2 tablespoons prepared horseradish, 1 tablespoon fresh lemon juice, 1 teaspoon Worcestershire sauce, and 1 teaspoon hot sauce.

Whisk until smooth and taste. Stir in 1 teaspoon smoked paprika (optional), 1 teaspoon coarse salt, and 1 teaspoon freshly ground black pepper. Adjust horseradish and hot sauce to your heat preference.

Cover and chill in the refrigerator for at least 30 minutes to allow flavors to meld. Sauce can be made up to 24 hours ahead; stir again before serving.

Shrimp

Prepare poaching liquid

Fill a large pot with 6 cups water. Add 1 tablespoon kosher salt, the halved lemon (squeeze halves into the water then add halves), 1 bay leaf, and 8 whole black peppercorns. Bring the mixture to a gentle boil over high heat.

Poach shrimp

Once the liquid reaches a rolling boil, reduce heat to maintain a lively simmer (not a vigorous boil). Add the shrimp in a single layer. Cook until the shrimp are opaque and just firm, about 2 to 3 minutes for large shrimp (16–20 count). Do not overcook — they will continue to firm slightly after removal.

Ice bath

Immediately transfer the cooked shrimp with a slotted spoon to a large bowl filled with 2 cups ice and cold water to stop the cooking. Let the shrimp chill in the ice bath for 3 to 5 minutes until completely cooled.

Dry and chill

Remove shrimp from the ice bath and place them on a paper towel–lined tray to drain and dry. Refrigerate uncovered for 10–15 minutes if you want them extra cold.

Serve

Arrange the chilled shrimp on a platter around a bowl of the chilled cocktail sauce or portion into individual glasses with sauce in the bottom and shrimp draped over the rim. Garnish with chopped fresh parsley and lemon wedges. Serve immediately.