Pat the whole chicken dry with paper towels and place it breast-side up on a stable cutting board.
Start with the legs: pull one leg away from the body to expose the hip joint and make a small slit through the skin just behind the drumstick to access the joint.
Pop the hip bone out of its socket by pulling the leg away, then cut through the joint with your knife or shears to remove the whole leg (thigh attached to drumstick).
Separate the drumstick from the thigh by cutting along the fat line between them; adjust the knife angle until you cut cleanly through the joint.
Remove the wings: with the breast-side up and wings extended, feel for the wing joint, then cut through the skin and joint (not the bone) while pulling the wing away for control.
To remove the breast: turn the chicken on its side and cut along the fat line at the side of the carcass. Use kitchen shears or a knife to snip through the ribcage and separate the breast from the backbone.
Debone the breast if desired: make a cut through the skin down the center of the breast to expose the breastbone, then use long, smooth strokes to cut the meat away from the bone until the breast is removed intact.
Repeat leg, wing, and breast removals on the other side to yield two breasts, two wings, two drumsticks, two thighs, and the backbone and breastbone.