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For relish, combine roasted pepper strips, olives, oil, thyme and 1/4 teaspoon pepper in a food processor. Cover and pulse several times until coarsely chopped. Cover and chill until ready to serve.
Combine onion, breadcrumbs, ketchup, horseradish, mustard, salt and the remaining 1/4 teaspoon pepper in a medium bowl. Add ground beef; mix well. Shape the meat mixture into four 3/4-inch-thick patties.
Grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until internal temperature of burgers reaches at least 160°F with an instant-read thermometer, turning once halfway through cooking. Top each patty with a cheese slice for the last 2 minutes of grilling.
Serve each burger on a toasted bun with fresh basil leaves and 1 tablespoon of the red pepper relish.