This half-chicken is marinated in a creamy dressing of oil, red wine vinegar, dried oregano and lots of garlic and lemon zest. The egg helps the marinade cling to the chicken while grilling so its flavorings penetrate the chicken deeply. Save time and buy half-chickens if you can. To ensure your fries are hot and ready when the chicken is, place them in a 500-degree oven while the chicken cooks on the grill (or, if using the oven for the chicken, put the fries on the rack below the chicken).
In a blender, combine the oil, vinegar, oregano, kosher salt, pepper and garlic. Using a vegetable peeler, remove the zest from both lemons in strips and add them to the blender; reserve the peeled lemons. Finally, add the egg and blend the mixture until smooth and emulsified, 20 to 30 seconds. Let the marinade stand while you break down the chicken.
Using kitchen shears, cut along each side of the chicken’s backbone and remove it; discard the bone or save for making stock later. Flip the chicken over and press on its breastbone to flatten it. Using a large knife or cleaver, cut through the breast bone to separate the chicken into two halves. Transfer the halves to a 9-by-13-inch baking dish and season the halves on both sides liberally with additional kosher salt and pepper.
Pour the marinade over the chicken and turn the halves to ensure they’re evenly coated. Arrange the halves skin side up in the dish, then cover the dish with plastic wrap and refrigerate the chicken for at least 1 hour or no longer than 4 hours. Remove the chicken from the refrigerator 20 minutes before you plan to cook it.
Heat your broiler to low, or arrange a rack 6 inches below the heating element and switch the broiler to high. Line a rimmed baking sheet with two layers of foil. Lift the chicken halves from their marinade and, without wiping off any of the marinade, transfer the halves, still skin side up, to the prepared baking sheet. Season once more with kosher salt and pepper. Cut the reserved lemons in half and arrange the halves cut side up around the chicken.
Broil the chicken halves and lemon halves on the sheet, rotating the sheet every 5 minutes or as needed, until the skin on both halves is bubbling, golden brown and crisp, about 15 minutes. Turn off the broiler and heat the oven to 500 degrees. Place the chicken in the oven and roast, rotating the sheet once halfway through, until the chicken halves are golden brown all over and cooked to an internal temperature of 160 degrees, 25 to 30 minutes.
If grilling the chicken halves, prepare a gas or charcoal grill for both direct and indirect grilling. Place the chicken halves skin side down and the lemon halves cut side down on the direct side of the grill and cook, turning as needed, until the chicken’s skin is golden brown and crisp, 10 to 15 minutes. Flip the chicken halves over and place on the indirect side of the grill; repeat with the lemon halves. Close the grill and continue cooking until the chicken is cooked through and an instant-read thermometer inserted between the thigh and drumstick reaches 160 degrees.
Remove the chicken halves from the oven or grill and transfer to a large serving platter lined with fries, cooked rice or warm pita. Squeeze the roasted lemon halves all over the chicken, then sprinkle with flaky sea salt. Serve immediately.
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