Start your grill and get it set up for indirect heat. You want the temperature to hit about 300 to 350 degrees F, a nice medium heat. So depending on your grill time this out with the chicken prep.
Spray a rimmed cookie sheet with non stick spray.
Spatchcock your chicken.
Pop the chicken on the cookie sheet. Drizzle with olive oil, then season it. Add about half the barbecue sauce now.
Add the veggies to the pan as well.
A little tip for this, since you're doing indirect heat you want to sort of slide the chicken to the far side of the cookie sheet so it's the farthest from the flame. Depending on the size of your grill part of the cookie sheet will be over direct heat, and part over indirect.
Take a brick and cover it with aluminum foil. Set near the grill.
Pop the cookie sheet on the grill, on the indirect side. Set the brick on top of the chicken. This flattens the bird so it cooks evenly.
If you have a meat thermometer you can pop that in the thickest part of the thigh. But it's a tad hard to do with the brick on there so I tend to cook mine without a thermometer. If you're new to grilling get a really good digital read thermometer and check the thickest part of the thigh as you go, you want it to reach about 165 degrees. If you're not using one this should take between 60 to 90 minutes, depending on the size of your bird.
About halfway through the cooking process flip the bird over and sauce it again. You can leave a little to brush on at the end if you like, I usually do. Pop the brick back on.
Make sure you keep an eye on the veggies. You should turn them occasionally. I put them closest to the direct heat when it first hits the grill and then move them over as they start to soften. You can shift things around as you go with the big rimmed sheet which is something I love about doing it this way.
Right before it's finished flip it again and brush with any remaining sauce. Remove from grill.
Carve the bird, pop the veggies on platters. And try not to stab anybody in the hand with your fork.
Serve with a big old chicken loving smile!