Prepare the water: Fill a medium pot with water and bring it to a boil. Drop in the halved lemon and a handful of peppercorns.
Turn off the heat: Once the water hits a rolling boil, turn off the heat.
Cook the shrimp: Carefully add the shrimp to the hot water. Let them soak for about 3–5 minutes. Gently move them around in the pot.
Check doneness: The shrimp will turn white and slightly pink on the outside when fully cooked. If in doubt, cut one open to be sure.
Chill the shrimp: Drain the shrimp and place them on a paper towel to remove extra moisture. Then transfer them directly to the refrigerator to cool.
Make the sauce: In a mixing bowl, stir together the ketchup, horseradish, Worcestershire sauce, Calabrian chili peppers, lemon juice, and hot sauce. Taste and adjust any ingredients as needed.
Serve: Once the shrimp is chilled, serve them with the cocktail sauce on the side for dipping.