Make the cocktail sauce: In a bowl, whisk together the ketchup, prepared horseradish, lemon juice, Worcestershire sauce, hot sauce, and black pepper. Cover and chill.
Prepare an ice bath: Combine the ice cubes and cold water in a large bowl and set aside.
Build the poaching liquid: In a large pot, combine the water, kosher salt, the lemon (squeeze in the juice and add both halves), bay leaves, and black peppercorns. Bring to a boil, then reduce to a gentle simmer and let infuse for 5 minutes.
Poach the shrimp: Add the shrimp to the pot, return to a gentle simmer, and cook until just opaque and pink, 1–2 minutes (shrimp should curl into a loose C; internal temperature about 120–125°F).
Chill the shrimp: Using a slotted spoon, transfer shrimp immediately to the ice bath and cool completely, about 5 minutes.
Peel and devein: Drain the shrimp, peel (leaving the tail segment intact if desired), and remove any visible veins.
Serve: Arrange the chilled shrimp on a platter and serve with the cold cocktail sauce and lemon wedges.