In a medium bowl, combine cornmeal, the 2 tablespoons milk, the onion, garlic, salt, oregano, and lemon-pepper seasoning. Add ground lamb and ground beef; mix well using hands. Shape mixture into eight 1/4-inch-thick patties.
For filling, in a small bowl, combine feta cheese and the 1 tablespoon milk. Divide filling among four of the patties. Top with the remaining patties; press edges to seal.
In a small bowl, combine cumin and cayenne pepper. Sprinkle mixture over patties.
For a charcoal grill, place patties on the rack of an uncovered grill directly over medium coals. Grill for 12 to 16 minutes or until done (160 degrees F), turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover and grill as above.)
To serve, cut pita bread rounds in half crosswise. Serve burgers in pita halves with arugula. Makes 4 servings.