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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.
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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.
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Light the grill and lightly oil the grill rack.
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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.