Combine the mayonnaise and balsamic in a bowl, season. Add the carrot and apple, mix together, cover and leave to infuse at room temperature for approx. 30 mins.
Grill the burgers in a griddle pan for approx. 5 mins. on each side, keep warm. Place the bread (cut side down) in the same griddle pan and grill for approx. 3 mins., spread with mayonnaise. Spoon the coleslaw onto the bottom half of each burger bun. Place the edamame and quinoa burger on top of the coleslaw, arrange the lettuce on top. Cover with the top half of the burger bun.