Cut out the backbone of the chicken with kitchen sheers, and discard (or save to make chicken stock). Pry the chicken open, and lay bone down on a cutting board. Give it a few good whacks with the palm of your hand right down the middle to flatten the chicken.
Gently work the tips of your fingers under the skin. Rub half the pesto under the skin onto the flesh. Rub a bit of salt under the skin as well, then sprinkle the rest on the outside. Cover loosely, and keep in the fridge for at least 2 hours and up to 2 days.
Preheat grill to high (450°F), with zones for direct and indirect heat.
When fire is ready, place chicken skin side down on indirect heat. Rotate chicken clockwise 90 degrees after 15 minutes. Then flip, and brush with remaining pesto. Grill another 10-20 minutes, until chicken is cooked through, and skin is crisp.