In a large bowl, combine ground chicken, breadcrumbs, egg, garlic, onion, oregano, salt, and pepper. Very gently knead until just combined. Form mixture into 6 patties.
Preheat a grill to 375F. I would recommend grilling these burgers either on a greased grill pan or a sheet of aluminum foil (they tend to stick to the grill grates if you put them on directly) Grill chicken burgers, flipping once until golden-brown and meat registers 165F (about 8-10 minutes of cooking)
Assemble burgers by spreading a little tzatziki sauce on the bottom of a toasted bun. Place a lettuce leaf and several cucumbers on top, following with the chicken burgers, red onions, tomato slices, feta, and additional tzatziki sauce. Serve immediately and enjoy!