In a mixing bowl, combine the ground chicken, feta cheese, parsley, red onion, garlic, oregano, salt, and pepper. Mix until well combined, but do not overwork the meat.
Divide the mixture into four equal portions and shape each portion into a patty, about 1 inch thick.
Heat the olive oil in a large skillet over medium heat. Cook the patties for about 5-6 minutes on each side, or until they are cooked through and have an internal temperature of 165°F.
While the burgers are cooking, prepare the tzatziki sauce by mixing together the Greek yogurt, grated cucumber, garlic, dill, lemon juice, salt, and pepper in a bowl. Adjust seasoning to taste.
Toast the whole wheat burger buns if desired. Assemble the burgers by placing a patty on each bun, topping with a generous spoonful of tzatziki sauce and a handful of baby spinach or mixed greens.