Process the onion in a food processor for 10 to 15 seconds and turn out into the center of a tea towel. Gather up the ends of the towel and squeeze until almost all of the juice is removed. Discard juice.
In a large bowl, mix together the onions, egg, bread crumbs, garlic, mint,, rosemary, feta cheese, salt, and black pepper. Add the lamb and fold all the ingredients with the meat. Do not over mix.
Form the burgers – about 1/3 pound each, about 3.5 inches round. Refrigerate for 1 hour.
Cook on a well-greased grill, or skillet indoors, over medium heat (you may need to put a drip pan under the burgers to prevent flare-ups on the grill). Burgers are done when the internal temperature is about 165 degrees F.
Serve on hard rolls or pita bread with tzatziki sauce, chopped onion, tomatoes cucumber, and feta cheese.