Make burgers as normal. Light grill (with grill pan) or a grill pan on the stove top. Slice onion into 1/8" slices and brush with 1/2 tablespoon olive oil. Grill onions along side burgers until onions are lightly charring. Stack a few rings of onions on the burger and top with sliced cheese. Transfer burger to a low-heat spot on the grill and let sit as cheese melts.
Brush rye bread with remaining 1/2 tablespoon and place on grill. Let bread toast for 1-2 minutes on each side. Assemble burger and devour.