Mix the kangaroo, onion, caraway seed, thyme, egg, and lemon zest together in a bowl; season with salt and pepper. If mixture is too wet and sticky add breadcrumbs. Shape the mixture into 8 balls and flatten.
Heat the oil in a skillet over medium heat; cook the burgers in the hot oil until the burgers are cooked through and no longer pink in the middle, 5 to 7 minutes per side for well done. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C). Serve the burgers on the split buns with lettuce and tomato.