In a large mixing bowl, combine ground chicken, breadcrumbs, feta cheese, parsley, dill, garlic, oregano, salt, and pepper. Mix gently and let sit for a few minutes.
Shape the mixture into patties about 1 inch thick and place on a plate or baking sheet.
In a medium bowl, mix Greek yogurt, grated cucumber, garlic, lemon juice, and dill. Season with salt and pepper to taste. Refrigerate until ready to serve.
Heat a skillet or grill over medium heat and add olive oil. Cook the patties for 5-6 minutes on each side until they reach an internal temperature of 165°F.
Assemble the burgers by placing lettuce on the bottom half of each bun, adding the chicken patty, tzatziki sauce, tomatoes, and red onion. Top with the other half of the bun.