Peel and cut onion into four 1/4-inch-thick slices; refrigerate remaining onion for another use. Shape meat loosely into four 1/2-inch-thick patties; sprinkle with garlic powder, salt, and ground black pepper. Press one onion slice into the center of each patty and shape meat around onion until top of onion is flush with the surface of the meat patty.
For a charcoal grill, place meat, onion side up, on the rack of an uncovered grill directly over medium coals. Grill for 10 to 13 minutes or until meat is done (160 degrees F), turning once halfway through grilling. Top with cheese before the last minute of grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover and grill as above.) Brush kale leaves lightly with oil and add to grill the last 1 to 1 1/2 minutes of grilling.
To serve, place two kale leaves atop each bread slice. Top with a burger, onion side up.