Preheat the barbecue for medium-high heat and lightly oil the grate.
Mix the lamb mince, beef mince, mint, ginger, 1 teaspoon garlic, 1 teaspoon salt and pepper in a large bowl until just combined. Divide evenly into four portions and shape in to large patties. Set aside. Mix Greek yoghurt, lemon zest, 1 clove garlic and 1/2 teaspoon salt in a bowl; cover and refrigerate.
Cook the lamb and beef patties on the preheated grate until the burgers are cooked to your desired degree of doneness, 3 to 4 minutes per side for well done. Place the slices of onion and green tomato on the grill. Cook until lightly charred, about 1 minute on each side.
Spread yoghurt sauce over the sliced ciabatta rolls. Assemble each burger by placing the patty on the the roll and dividing the feta cheese slices over the patties. Top with a slice of tomato, onion, and 2 leaves of rocket, followed by the top half of the roll.