1. To prepare the patties, divide beef into 4 portions, lightly shaping each into a 1/2-inch-thick patty. Sprinkle evenly with salt. Cover and refrigerate 30 minutes.
2. To prepare horseradish spread, combine mayonnaise and next 3 ingredients (through ketchup) in a small bowl. Set aside.
3. To prepare relish, cook bacon in a large nonstick skillet over medium-low heat until crisp. Remove bacon from pan with a slotted spoon. Add onion to drippings in pan; cook 15 minutes or until golden brown. Combine bacon, onion mixture, chives, Worcestershire sauce, and pepper in a small bowl.
4. Heat a large cast-iron skillet over medium-high heat. Coat pan with cooking spray. Add patties; cook 2 minutes on each side or until desired degree of doneness. Spread 1 1/2 teaspoons horseradish spread on cut sides of each bun half. Top bottom half of each bun with 1 patty, 1/4 cup relish, 1 tomato slice, 1/4 cup lettuce, and top half of bun.