In a small bowl, stir together the mayonnaise and horseradish sauce. Add the pickle juice. Stir in one tablespoon of lemon juice. Chill until ready to serve.
Using a food processor, finely chop the shrimp, red onion, red bell pepper, green onions, and garlic. Then add the Old Bay seasoning and one tablespoon of lemon juice. Pulse well to combine and remove any large chunks.
Transfer the ground shrimp to a bowl and add the breadcrumbs and egg. Stir to combine. Form the shrimp mixture into quarter-pound patties, about a half-inch thick.
In a skillet over medium-high heat, add the sesame oil and heat until shimmering. Add the shrimp burgers in batches and cook on both sides for 1-2 minutes, or until the shrimp is cooked through (about three minutes total).
Serve the shrimp burgers on burger buns and top with avocado, tomatoes, radicchio, and of course, the horseradish tartar sauce. Serve hot and enjoy!