Combine all burger ingredients except oil. Form into patties and place on a lined tray, refrigerate for 30 minutes to set. Mix mayonnaise, chilli paste and sugar in a small dish.
Brush patties on both sides with oil and cook over medium heat – in a heavy-based skillet or on a pre-heated barbecue – for 3-4 minutes per side or until cooked through. Transfer to a clean plate and cover loosely with foil to keep warm.
Place lettuce on plates, top with cucumber, carrot and burgers. Top with mayo and kimchi. Serve with steamed rice on the side.
For a classic burger with a Korean twist, stack patties, lettuce and toppings on toasted buns.