In a large mixing bowl, combine the ground chicken, crumbled feta, chopped spinach, red onion, garlic, oregano, lemon zest, salt, and pepper.
Form the mixture into four equal-sized patties, approximately 3/4-inch thick.
Heat the olive oil in a non-stick skillet or grill pan over medium-high heat.
Place the patties in the pan and cook for 5 to 6 minutes per side.
Verify the internal temperature has reached 165°F (74°C) using an instant-read thermometer.
Remove the patties from heat and let rest for 2 minutes.
Toast the buns lightly and spread a generous tablespoon of tzatziki on each.
Assemble the burgers by placing the chicken patty on the bun and topping with cucumber and tomato slices.