Bring a large pot of water to a gentle simmer with 1 tablespoon of the kosher salt. While the water heats, stir the ketchup, horseradish, lime juice, Worcestershire sauce, and 1/4 teaspoon salt together in a small bowl.
Cut the cucumbers into 1-inch rounds. Scoop out the centers with a melon baller, leaving a 1/4-inch base, and set the cups aside.
Add the shrimp to the simmering water and poach until pink and opaque, 2 to 3 minutes. Transfer to a bowl of ice water for 3 minutes, then drain and pat dry.
Spoon about 1 teaspoon of the cocktail sauce into each cucumber cup and top with one shrimp.
Finish with diced avocado and chives. Serve immediately, or refrigerate for up to 2 hours.