Cut the carrots into 2 1/2-inch long segments. Cut the onions into wedges. Place on a sheet pan, drizzle with 1/4 cup of the oil and a pinch of salt and pepper.
Next, spatchcock the chicken by placing on a board. Place the chicken breast-side down and locate the backbone. Cut down both sides of the bone with kitchen shears and remove the backbone. Discard this or use to make stock. Flip the chicken over and press firmly with the palm of your hand on the center of the breast. The breastbone should pop and the chicken should sit relatively flat.
Season the chicken all over with the remaining oil, 2 tablespoons salt and 2 teaspoons pepper. Place on top of the vegetables on the sheet pan. Place in the oven and roast for 25 minutes.
Remove the pan from the oven, sprinkle over the garlic and toss with the carrots and onions. Brush the chicken all over with the BBQ sauce, return the pan to the oven and roast for 10 minutes.
Remove from the oven, brush with more BBQ sauce and return to the oven for another 10 minutes.
Remove from the oven, brush on more sauce and return it to the oven for a final 10 minutes, cooking until the chicken registers 165 degrees F when a meat thermometer is inserted in the thigh.
Remove from the oven and rest for 10 minutes, covered loosely in foil.
Transfer the chicken to a cutting board and cut into 4 or 6 pieces. Put on a platter and surround it with the vegetables. Serve with additional BBQ sauce for dipping.