Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Mix chicken, bread crumbs, egg, lemon juice, sun-dried tomatoes, basil, oregano, salt, and pepper together in a bowl; form mixture into 5 patties. Divide feta cheese into 5 equal portions; place a portion of feta cheese atop each patty and fold chicken mixture around cheese to secure it in the center.

Cook patties on the preheated grill until no longer pink on the inside and juices run clear, 5 to 7 minutes per side. An instant-read thermometer inserted into the center should read 165 degrees F (75 degrees C).