Salt zucchini and eggplant; let sit 30 minutes on a rack with paper towels underneath.
Meanwhile, place ground chicken or turkey in a medium-sized bowl. Add onions, breadcrumbs, egg, salt, pepper, parsley and mint, and knead with your hands, until well-combined. Chill 30 minutes.
Pat excess water from eggplant and zucchini with a dry paper towel. Brush both sides generously with olive oil; sprinkle with dried oregano and pepper. Set on a platter.
Shape meat mixture into 4–6 long, flat meatballs. Set on a platter.
Heat grill to medium heat; oil well.
Place meatballs on grill; cook 8–10 minutes, turning once.
When meatballs are half done, place eggplant and zucchini on grill. Cook 2–3 minutes on each side.
Place meat and vegetables on a platter with pita bread, lettuce, tomatoes, cheese and a bowl of tzatziki. Let guests make their own sandwiches.