Step 1: Peel and devein the shrimp, leaving the tails on. Rinse under cold water and pat dry with paper towels. Set aside.
Step 2: In a large pot, combine water, 1 tablespoon of salt, black peppercorns, bay leaves, and lemon slices. Bring the mixture to a rolling boil over high heat.
Step 3: Add the prepared shrimp to the boiling liquid. Stir gently. Cook for 1-3 minutes, or until the shrimp turn opaque pink and curl into a 'C' shape. Do not overcook.
Step 4: Immediately transfer the cooked shrimp to a large bowl of ice water using a slotted spoon. Let them chill for at least 5 minutes to stop the cooking process and maintain tenderness. Drain thoroughly and pat completely dry. Refrigerate until ready to serve.
Step 5: While the shrimp are chilling, prepare the cocktail sauce. In a medium bowl, whisk together the ketchup (or chili sauce), prepared horseradish, fresh lemon juice, Worcestershire sauce, and hot sauce (if using).
Step 6: Season the cocktail sauce with salt and black pepper to taste. Mix well. Taste and adjust horseradish or lemon juice as desired.
Step 7: Cover the cocktail sauce and refrigerate for at least 30 minutes to allow the flavors to meld. The sauce can be made several days in advance.
Step 8: To serve, arrange the chilled shrimp around the rim of a serving platter or in individual glass goblets, with the tails facing outwards. Spoon the cocktail sauce into a central dipping bowl or smaller individual bowls.
Step 9: Garnish with fresh lemon wedges and chopped parsley or dill, if desired. Serve immediately and enjoy your homemade shrimp cocktail recipe!