Over medium/high heat in a non-stick skillet, add onions, sugar and a pinch of salt and pepper.
Reduce the heat to medium and cook onions and sugar, stirring occasionally, until the onions are golden brown, around 15-20 minutes.
Stir in vinegar and cook another minute. Remove from heat.
Combine the mayonnaise, mustard and basil in a small bowl. Squeeze the lemon into mixture; blend and reserve.
Heat grill. Form the ground beef into 4 (4” diameter) patties, sprinkle with salt and pepper on each side.
Grill burgers (about 5-6 minutes per side).
Top each burger with 1 slice of cheese. Grill for another 1 or 2 minutes until cheese is melted.
Cut roll in half, put on grill face down, grill until desired doneness.
Assemble burger: Start with romaine lettuce, add 1 slice of tomato, grilled burger and top with caramelized onions. Spread Basil Mayonnaise on top of brioche roll. Serve immediately.