Combine first 4 ingredients and 1/4 teaspoon salt in the bowl of a mini food processor; process until smooth.
Combine ground beef and red wine in a medium bowl; mix gently. Divide mixture into 4 equal portions; gently shape each portion into a 1/2-inch-thick patty. Sprinkle patties evenly with 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Heat a grill or grill pan over medium-high heat; coat with cooking spray. Place patties on grill, seasoned sides down, and sprinkle with remaining 1/4 teaspoon each salt and pepper. Grill 3 minutes; flip patties and grill 4 more minutes or until desired degree of doneness. Top each patty with 1 slice cheese during last 30 seconds of cooking. Remove burgers from grill; cover loosely with foil. Grill onion slices 3 minutes on each side or until softened and lightly charred.
Spread 3 tablespoons horseradish butter evenly over cut sides of rolls; grill, cut sides down, 1 minute or until lightly toasted. Remove rolls from grill, and spread remaining horseradish butter over roll bottoms; top with tomato slices, patties, onion slices, arugula, and bun tops.