raw shell-on shrimp (extra-jumbo, 16 to 20 per pound), thawed if frozen
large white onion
water
kosher salt
white wine vinegar
black peppercorns
medium lemons
ketchup
Worcestershire sauce
prepared horseradish, plus more as needed
freshly ground black pepper
vinegar-based hot sauce, such as Tabasco
yuzu kosho, sambal oelek, or other chili paste, plus more as needed (optional)
Lemon wedges, for serving (optional)
Rinse 2 pounds raw extra-jumbo shrimp under cool water. Devein with kitchen shears, but do not peel.
Peel and quarter 1 large white onion and place in a medium pot. Add 8 cups water, 2 tablespoons kosher salt, 2 tablespoons white wine vinegar, and 1 tablespoon black peppercorns. Bring to a boil over high heat. Add the shrimp and remove from the heat. Let sit until just opaque, pink, and cooked through, 1 1/2 to 2 minutes.
Drain or remove with a slotted spoon and spread on a baking sheet in a single layer; discard the onion and peppercorns. Refrigerate until chilled through, about 25 minutes. Meanwhile, make the cocktail sauce.
Juice 1 of the medium lemons into a medium bowl until you have 2 tablespoons. Add 1 1/2 cups ketchup, 2 tablespoons Worcestershire sauce, 1 tablespoon prepared horseradish, 1 teaspoon black pepper, 1/2 to 1 teaspoon vinegar-based hot sauce, and 1 tablespoon yuzu kosho, sambal oelek, or other chili paste, if using. Stir to combine. Taste and add more horseradish and/or chili paste, if desired.
Peel the shrimp, leaving the tails on. Halve the remaining 1 medium lemon and squeeze the juice over the shrimp. Divide the cocktail sauce between 8 martini or coupe glasses, then hang 5 shrimp over the edge of each glass. Alternatively, fill a lipped platter or shallow serving bowl with ice, then place the shrimp on the ice with a bowl of cocktail sauce alongside. Serve with lemon wedges for squeezing, if desired. Set out a small bowl for discarded shrimp tails.