Combine bulgur and 1/3 cup boiling water in a 2-cup glass measure. Let stand 10 minutes, then microwave until liquid is absorbed, 2 to 3 minutes. Let cool.
Meanwhile, cook chopped onion in 1 tbsp. oil in a large frying pan over medium heat until deep golden, stirring often, 10 to 15 minutes. Stir in garlic, scrape into a large bowl, and let cool. Brush sliced onion with remaining 1 tbsp. oil and set aside. In a bowl, combine mayonnaise, lemon zest, 1/2 tsp. rosemary, and 1/4 tsp. pepper; chill.
Heat grill to medium (350° to 450°). Whirl cherries in a food processor until they're finely ground and form a ball. Add them to chopped onion with remaining 2 tsp. rosemary and 1/2 tsp. pepper, the salt, and bulgur; blend well with your fingers. Add ground beef and mix gently but thoroughly. Shape into 4 patties, 1/2 in. thick.
Grill onion slices, turning occasionally, until well browned, 10 to 15 minutes. Grill burgers, turning once with a wide spatula, until they no longer stick to grate and grill marks appear, 10 to 12 minutes total; about 2 minutes before burgers are done, top them with brie and grill the buns.
Spread buns with mayonnaise; add lettuce, burgers, and onions.