Make the cocktail sauce first so it has time to chill and the flavors settle. Stir together the ketchup, horseradish, lemon juice, Worcestershire sauce, and hot sauce in a small bowl. Taste it and add more horseradish or hot sauce until it has the kick you want. Cover and refrigerate until ready to serve.

Fill a large saucepan with about 6 cups of water. Add the salt, peppercorns, smashed garlic, and a squeeze from one lemon half. Bring to a gentle boil over medium-high heat.

While the water heats, set up your ice bath. Fill a large bowl with ice and cold water and set it near the stove.

Add the shrimp to the boiling water in one layer if possible. Stir once. Cook for 2 to 3 minutes, watching closely. The shrimp are done the moment they are fully pink on the outside and opaque all the way through. Do not walk away. Overcooked shrimp turn rubbery and cannot be saved.

Use a slotted spoon to transfer the shrimp immediately to the ice bath. Let them sit for 3 minutes until completely cold, then drain and pat dry with paper towels.

To serve, fill a large platter or shallow bowl with crushed ice. Pour the cocktail sauce into a small serving bowl and nestle it in the center of the ice. Arrange the shrimp around the bowl, tails pointing outward. Tuck lemon wedges from the second lemon half around the platter for guests to squeeze over their shrimp.