Remove the packets of giblets from the body of the chickens and set aside for another use. Remove and discard the fat just inside the body and neck cavities of the chickens. Rinse the chickens, inside and out, under cold running water and then drain and blot dry, inside and out, with paper towels.
Spatchcock the chickens: Place the chicken on its breast. Using poultry shears, start at the neck and cut lengthwise down one side of the backbone to the tail. Next, repeat that cut along the otherside of the backbone. Remove the backbone and save it for stock.
Now, fold the chicken open, like a book, with the skin side down. Use a paring knife to cut along each side of the breastbone. Run your thumbs along both sides of the breastbone and pull the white cartilage out.
Turn the skin side up and trim the wingtips and any loose skin. The chicken is now spatchcocked. Place the chickens in a nonreactive roasting pan.
Prepare the pesto: Place the dill, basil, garlic, walnuts, cheese and lemon zest in a food processor and process until very finely chopped. With the machine runing, add the oil, lemon juice, salt, pepper and 3/4 cup water. Taste for seasoning, adding salt and pepper as necessary; the pesto shuld be highly seasoned. Set 1/2 cup pesto aside. pour the remaining pesto over the chickens, spreading it on the meat with a rubber spatula. Turn the birds so you coat both sides. Let the chickens marinate in the refrigerator, covered, for as little as 2 hours or as long as overnight, turning once or twice to coat with marinade; the longer the chickens marinate, the richer the flavor will be.
Set up the grill for direct grilling and preheat to medium. When ready to cook, brush and oil the grate. Arrange the chickens on the grate, skin=side down. Grill for 12 to 15 minutes per side, turning once with tongs and a spatula. the chickens can be a little awkward to turn; you'll need to use both utensils. When you turn the chickens, spread the reserved 1/2 cup pesto on top. You'll probably get flare-ups as the melting fat hits the fire. Keep the birds moving to dodge the flames. If the skin browns to much, lower the heat or move the chickens to a cooler section of the grill. The chickens are done when an instant-read meat thermometer inserted into the thickest part of a thigh (but not touching the bone) registers about 180 degrees F.
Transfer the chickens to a plater, let rest for a few minutes, garnish with the dill sprigs and lemon wedges, and serve.
Tip 1: Any bird can be spatchcocked and grilled in this fashion, from qual to turkeys.
Tip 2: Feel free to make the pesto with rosemary (1 cup of leaves) or another fresh herb.