For relish, combine red sweet pepper strips, ripe olives, olive oil, thyme, and 1/4 teaspoon pepper in a food processor bowl. Cover and process with several on-off turns until coarsely chopped. Cover and chill until ready to serve.
Combine onion, bread crumbs, catsup, horseradish, mustard, salt, and 1/4 teaspoon pepper in a medium mixing 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 degrees F with an instant-read thermometer, turning once halfway through cooking. Top each patty with a cheese slice the last 2 minutes of grilling.
Serve each burger on a toasted whole wheat bun with fresh basil leaves and 1 tablespoon red sweet pepper relish. Makes 4 servings.