Preheat oven to 400° F.
Set the whole chicken upright on its legs, with the backbone facing towards you. Using a very sharp knife, cut out the backbone: grasp one side of the chicken with your free hand and place the tip of the knife near the neck with the other hand and push downward, pressing the side of the knife against the outside edge of the backbone. Push the knife downward, keeping the side against the backbone as firmly as possible. Cut all the way from the top of the neck, down through the body all the way to the joint where the leg meets the backbone. Cut through or around the leg joint. When one side is detached, repeat on the other side. Set the backbone aside for making stock, if desired.
Now, open the chicken from the backside, as if it were a book. The t-shaped breastbone is in the upper third of the body. With your knife, give the breast bone a few “taps” to break the top layer of cartilage, then carefully snap the wings and breasts back to expose the sternum, which is a beak-shaped piece of cartilege between the breasts. You can either remove this piece with your fingers if your hands are pretty strong, or by carefully cutting it out. You can also just break it in half with another few more thwacks of your knife’s tip. However you remove it, the bird should now lay more or less flat, skin-side down.
Take the wing tips off if you’d like, they tend to overcook in this recipe but it’s not a big deal.
Turn the chicken over so it’s now skin-side up and place on a high-sided baking sheet lined with parchment paper or a silicone mat, if you have either. If you don’t, it’s just a little more scrubbing at the end. Push the chicken body down as flat as possible. With the sternum removed it should removed it should lay fairly flat, although the ribs are still intact. Turn legs “wrong-side out”, so the thickest parts of the thighs are against the body and the foot joints point outward. Tuck the wings underneath if you’ve left them on.
Combine most of the oil with salt, pepper and paprika and rub it on top and underneath the skin. Surround the chicken with veggies and drizzle them with remaining oil.
Make sure the baking rack is in the center of the oven. Pull it out and place the baking sheet with the chicken and veggies on the rack. Carefully pour the stock onto the baking sheet over the vegetables., tipping the sheet if you need to so the liquid covers almost all of the baking sheet. You should have a thin pool of liquid at the bottom of the pan. Carefully push the rack with the sheet pan on it back into the oven.
Bake chicken until a thermometer inserted into the thickest part of the thigh registers 165° F., about 40 to 50 minutes, giving the pan a turn once. If desired, turn the baking setting to broil for the last 5 minutes to create a very crisp skin.
Remove the chicken to a table set with a trivet, towels or a heatproof surface. Let the chicken cool for 5-10 minutes. Cut into quarters or eighths and serve with vegetables.