Preheat an outdoor grill for medium-high heat, and lightly oil the grate.
Mix the chicken, bread crumbs, egg, lemon juice, sun-dried tomatoes, basil, oregano, salt, and pepper together in a bowl; form the mixture into 4 thin patties.
Divide the feta cheese between the 4 patties; place a portion of the feta cheese atop each patty and fold the chicken mixture around the cheese so that the cheese is in the center and seal well.
Cook on the preheated grill until no longer pink on the inside and the juices run clear, about 5 minutes per side. An instant-read thermometer inserted into the center should read 165 degrees F (75 degrees C).