Preheat your oven to 425°.
Find your heaviest-bottomed roasting pan or shallowest oven-friendly dish. You'll also need a large skillet.
PREP THE CHICKEN Rinse your chicken under cold running water. Allow to air dry or pat it dry with good quality paper towels. Use the cheap ones and eat paper. 1) Cut through the skin between the leg and body. Twist the leg down until the joint pops out of the socket. 2) Use your chef's knife to remove the leg by cutting through the exposed joint. Repeat on other side. 3) Remove the wings. 4) Cut each wing crosswise into two pieces. 5) Hold the chicken vertically with the tail end up. Cut through the skin and cartilage to remove the backbone. 6) To split the breast, cut through both sides of the breastbone until you hit the sternum. Cut through the bone.
Place the chicken in a bowl. Lightly spray the chicken with cooking spray; toss with the salt, sugar, pepper, and thyme leaves. Put it in the refrigerator until it's time to cook.
Clean up after yourself. Cross-contamination is a very bad thing indeed.
CHOP YOUR VEGGIES
Set up a vegetable scrap container for the imperfect work you're about to do. You can compost the scraps or discard them. Grab a medium bowl for the prepped veggies, too.
It's time to apply the knife skills lesson you learned. Start with dicing the celery. It's the easiest.
Now, dice the onion.
Great! Now that your confidence is up, finish with the most challenging of the three veggies to dice nicely--the carrots.
Thinly slice the lemon. Keep the sliced lemon separate from the diced veggies.
BROWN THE BIRD Heat a large skillet over medium-high heat. Add 1 tablespoon of oil and swirl. If the oil smokes, you've gone too far. It happens. Back off the heat and lift the pan from the cooking area to let the smoke subside. Return to calm.
Add half the chicken to the pan, skin side down. Cook for 4 minutes until the chicken skin looks like the best, crustiest bread you've ever seen. Turn and cook 3 minutes more.
Remove from pan. Repeat procedure with remaining oil and chicken.
Drain oil from pan. It's exhausted.
Add the celery, onion, carrots, and thyme; cook 4 minutes or until lightly browned, scraping pan to remove brown bits. As the vegetables collapse and release their liquid, they're deglazing the pan, and that spells delicious.
EVERYTHING INTO THE OVEN On the roasting pan, make a uniform bed of the cooked vegetables, and lay all of the chicken pieces on top neatly. Like an artist, tuck the lemon slices and the bay leaves into the pockets between the chicken pieces. Nearly bury the bay leaves. You want those to fall into the juices as the chicken roasts.
Roast the chicken at 425° for just over an hour, checking after 30 minutes, until the internal temperature (at the bone) of the thickest piece of chicken is 155°. This is the point where many think the juices should run fully clear. Not yet. We have to account for carry-over cooking. Remove the chicken from the oven.
MAKE A PAN SAUCE After 10 minutes, grab your smallest saucepan and carefully tilt the roasting pan to drain off the juices. Add the sherry vinegar to the pan; bring it to a boil over medium heat. Let it reduce until nearly syrupy, about 6 minutes or until reduced to 1/2 cup. Turn off the heat and swirl in the butter as you walk away with the pot in your hand, looking like a pro.
Grab a platter, tongs, a serving spoon, and a large soup spoon. Move the chicken out of the way of the vegetables for a moment so you can mound your mirepoix on the platter--front and center. We want everyone to know about your Mad Skills.
Now, place the chicken pieces on and around the veggies. Don't be precise; be deliberate. There's a difference. The layout should look natural, comforting, and casual.
Pour your beautifully rustic sauce over the top and all around.