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Make the cocktail sauce: In a medium bowl, mix ketchup, horseradish, black pepper, lemon juice, hot sauce, Worcestershire, and salt. Put in small ramekins for dipping.
Make the mignonette: In a medium bowl, mix shallot, vinegar, pepper, and tarragon, then pour into ramekins for dipping.
Cook the crab: Bring a medium pot of water to a full boil. Add celery, onion, garlic, lemon, and black peppercorns. Using a pasta strainer, place the crab claws into the boiling pot for 6 to 8 minutes, then reduce heat to a low simmer. You will notice a dramatic change in the crab claws as they turn bright orange.
While the crab is poaching, fill a medium bowl with ice water. When the crab is done, transfer it to the ice bath to halt the cooking process. Once the crab has completely cooled, drain the water and pat the claws dry with a kitchen towel. Crack the claws and keep cool on ice in the refrigerator until ready to serve.
Shuck the oysters.
To serve: Place oysters, crab claws, and sauce ramekins on a tray covered with crushed ice, and enjoy with friends.