Step 1

Shrimp: Cover a rimmed baking sheet with ice (to later cool down shrimp). In a large stockpot combine 12 cups water, carrot, celery, thyme, 1 tablespoon salt, and peppercorns. Cut one lemon in half; squeeze half the juice into the water and add the rind. Cut remaining 1 1/2 lemons into small wedges for serving; set aside.

Step 2

Bring water to a boil. Add shrimp, cover; remove from heat. Let stand until shrimp are bright pink and curled; 3 to 5 minutes. Drain and transfer to prepared tray of ice to cool completely. Once cool, shrimp can be refrigerated in an airtight container up to 2 days.

Step 3

Cocktail Sauce: Meanwhile, whisk together all ingredients until combined. (Cocktail sauce can be made ahead; refrigerate in an airtight container up to 1 week.)

Step 4

To Serve: Peel shrimp, leaving tails intact if desired. Using a paring knife, make a shallow cut along backs of shrimp to remove vein. Arrange shrimp neatly on a platter, shingling them slightly to make a pattern, and serve chilled with cocktail sauce and reserved lemon wedges on the side.