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Soak eight wooden skewers in water, 15 minutes. Heat a grill or grill pan to medium-high. Clean and lightly oil hot grill. Soak sliced onion in warm water, 10 minutes. Using your hands, gently combine grated onion, chicken, garlic, chopped mint, feta, and 1/2 teaspoon each salt and pepper. Divide mixture into 8 portions, form around skewers, and brush with 1 tablespoon oil. Grill until cooked through, about 12 minutes, flipping once.
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Drain sliced onion and toss with 1 tablespoon oil, whole mint leaves, tomatoes, and vinegar; season with salt and pepper. Serve kebabs in lettuce leaves, topped with onion salad.