Mix all of the meat ingredients together (minus the feta) and form half of the beef mixture into 7 patties. Put a large chunk of feta cheese on top of each half of the patties. Put the remaining beef mixture flattened on top of each patty, covering the feta cheese. Grill the burgers until cooked through on the grill (or broil or use an indoor grill). Make the cucumber sauce by mixing all of the ingredients together until smooth. Serve the burgers with lettuce, tomatoes and the cucumber sauce on top.