Preheat a grill to high heat. Add ½ cup of water to panko breadcrumbs and stir until fully soaked.
Add ground beef to a large bowl and make a well in the middle. Add egg, onion, soaked breadcrumbs, oregano, mint, salt and pepper. Use your hands to mix all the ingredients together until well incorporated. Form the beef into 6 even sized balls. Stuff each burger with feta by using your finger to make a deep hole in the center of each ball. Stuff about 1 tsp inside each and pinch close. Place the ball on a flat surface and then press it down to form a patty.
Place burgers on the grill for about 8 minutes. Serve on a bun with additional feta cheese on top and sliced grape tomatoes.
