Heat an outdoor gas grill on medium for 20 to 30 minutes or so, meanwhile, prepare the meat mixture, pita (or buns) and fixings for the burgers.
Add the ground lamb (or mix of lamb and beef) to a mixing bowl. Add grated onions, garlic, fresh herbs, oregano, cumin, paprika and cayenne. Season with kosher salt and pepper. Add a generous drizzle of extra virgin olive oil, and mix until everything is well combined.
Divide the meat mixture into 8 equal balls. Using both hands, work meat into smooth spheres, then lightly press to form into patties. Arrange the patties on a large tray or dish, and with the tip of your thumb, press lightly to form a shallow, somewhat wide, depression into the center of each patty. (see cook's tip #2 below) Cover and refrigerate until grill is ready.
Lightly oil grill grates and arrange the burger patties on top. Grill over medium heat, covered, for 4 to 5 minutes on each side, turning once midway through cooking (internal temperature for a medium burger should register 160 degrees F. Adjust cooking time according to your desired doneness.)
Allow the lamb burgers to rest for 5 to 10 minutes before serving. Serve in pita (or buns), with tztaziki sauce, tomatoes, cucumbers, peppers, onions. If you like add kalamata olives and a sprinkle of feta.