Place chicken breast side down on cutting board. Using kitchen shears, cut through bones on either side of backbone; discard backbone. Trim chicken of excess fat and skin. Flip chicken over and press on breast-bone to flatten. Pound breasts with meat pounder to even thickness.

Grate 2 teaspoons zest from lemon and mix with 2 teaspoons salt and 1 teaspoon pepper in bowl. Pat chickens dry with paper towels and, using your fingers or handle of wooden spoon, gently loosen skin covering breasts and thighs. Rub zest mixture under skin, then season exterior of chicken with salt and pepper. Tuck wingtips behind breasts and transfer chickens to prepared rack. Refrigerate, uncovered, for at least 1 hour.

Turn all grill burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn other burners to low. (Adjust primary burner as needed to maintain grill temperature of 350 to 375°F.)

Oil grate. Halve remaining lemons and place cut side down on hotter side of grill. Place chickens skin side down on cooler side, with legs pointing toward fire. Cover.

Grill lemons until deep brown and caramelized, 5 to 8 minutes; transfer to bowl. Continue to grill chickens, covered, until breasts register 160°F and thighs register 175°F, 40 to 50 minutes longer. Slide chickens to hotter side of grill and cook, uncovered, until skin is well browned, 2 to 4 minutes. Transfer chickens to carving board skin side up, tent loosely with aluminum foil, and let rest for 15 minutes.

Squeeze 1/3 cup juice from grilled lemons into bowl. (Cut remaining into edges for serving.) Mash garlic and 1/2 teaspoon salt into paste and add to bowl with lemon juice. Whisk in parsley, mustard, sugar, and 1/2 teaspoon pepper. Slowly whisk in oil until emulsified.

Carve chickens, and pour 1/3 cup vinaigrette over. Serve, passing remaining vinaigrette and grilled lemon wedges separately.