Peel and wash the onions then grate along the fine-holed side of a manual cheese grater. Alternatively, pulverize the onions in a food processor. Place the grated or pulverized onions in a fine mesh strainer or in a cheese cloth and press or squeeze to extract their juices. Two grated or pulverized white onions yield about a cup of juice. Mix the onion juice with the olive oil and a little salt. Set aside.

In a large bowl, combine the ground chuck and lamb, onion, cumin, paprika, cayenne, parsley, salt, pepper, and olive oil and knead well. Cover with plastic wrap and refrigerate for one hour.

Light the grill and lightly oil the grill rack.

Rub the palms of your hands with a little oil (it helps keep the burgers from sticking on the grill). Shape the ground meat mixture into 8 equal-sized burgers. Place on the grill and cook, turning and brushing with the onion juice marinade, for about 4-5 minutes per side, or until medium. Remove and set aside, covered, to keep warm. Lightly grill the pita halves. Serve the burgers with the pita on a platter or on individual plates, and spoon remaining onion juice marinade over the cooked burgers for additional flavor.