Place the chicken in a 9-by-13-inch glass baking dish. Whisk together the lemonade, garlic, mustard, oregano, dried onion, cumin, 1 tablespoon salt and a few good grinds of pepper in a large liquid measuring cup until combined. Pour over the chicken, making sure all the pieces are fully submerged. Wrap tightly with plastic wrap, then refrigerate at least 4 hours or preferably 24 hours.
Prepare a grill for medium heat, and lightly oil the grill grates.
Remove the chicken from the marinade; reserve 2 cups of marinade and discard the rest. Place the chicken on a paper towel-lined baking sheet and pat dry. Sprinkle the chicken on both sides liberally with salt and pepper and arrange the pieces skin-side down on the grill, leaving some space between each. Cook until the skin is lightly charred and releases easily from the grill, about 2 minutes. Rotate the chicken pieces about 90 degrees so the skin won't burn and cook another 2 minutes. Flip the chicken skin-side up. Cover the grill and cook until an instant-read thermometer inserted into the thickest part of each thigh reads 165 degrees F, 20 to 25 minutes more. If the chicken pieces are getting too dark in spots, move them around the grill to cook evenly. Transfer to a platter and let rest for about 5 minutes before serving.
Meanwhile, transfer the reserved marinade to a small saucepan and place on the still-hot grill. Bring to a boil, then continue to cook until it has thickened slightly and reduced by half, about 20 minutes.
Finely grate the zest from 1 of the lemons, then sprinkle over the chicken. Squeeze the juice from half of the zested lemon (about 1 tablespoon) into the sauce and stir to combine. Transfer to a small bowl for serving. Cut the remaining 2 lemons in half crosswise.
Place the lemon halves on the grill, flesh-side down, and grill until lightly charred, 1 to 2 minutes. Serve alongside the chicken and sauce.
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