Step 1: Prepare a charcoal grill for two-zone cooking over medium-high heat by pouring the coals onto one half of the grill. For a gas grill, heat all of the burners to high, then turn off one of the end burners before cooking. (See Tips.)
Step 2: While the grill’s heating, pat the chicken dry. If the breasts are uneven in their thickness, pound until even with a heavy skillet or meat mallet. Set aside to air-dry. In a rimmed dish or wide, shallow bowl, stir together 1/2 cup olive oil, parsley, olives, olive brine and chile. Using the small holes on a box grater, grate 2 teaspoons zest from the lemon. Add to the bowl, squeeze in half the lemon (about 1 1/2 tablespoons juice), and season to taste with salt and pepper. Cut the remaining lemon half into wedges for serving.
Step 3: When you’re ready to grill, season the chicken with 1 teaspoon salt, then lightly coat with olive oil. Take the chicken, sauce, tongs and a tightly folded paper towel soaked in olive oil to the grill. Clean the grates with a grill brush, then oil the grates with the paper towel. Grill the chicken until well browned on one side and it easily releases from the grates, 4 to 6 minutes. (If flare-ups occur, move the chicken to an area of the grill where there are no flames underneath. For a gas grill, close the lid between flips, listening and keeping an eye out for flare-ups.) Flip and cook until the breasts register 155 degrees in the thickest part and the thighs register 165 degrees, 2 to 3 minutes for breasts and 4 to 5 minutes for thighs.
Step 4: Transfer the chicken to the sauce and turn to coat. Let rest for at least 5 minutes and up to 30. Season to taste with salt, pepper and a squeeze of lemon.
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