Whisk the brown sugar, smoked paprika, garlic powder, cumin and salt together.
Cut the backbone out of the chicken using poultry shears. Flip the chicken so that the breast side is up, and press down on the breastbone to flatten the chicken.
Brush both sides of the chicken with oil. Rub the chicken with the spice blend, using about 2⁄3 of the spices on the breast side and the remaining 1⁄3 on the inside of the chicken.
Allow the chicken to rest at room temperature for 30 minutes to an hour.
Lay a piece of aluminum foil over the grates of a grill. Brush the foil with oil, and then preheat the grill to medium.
Put the chicken, breast-side down, on the hot foil. Place the foil-wrapped bricks over the chicken so that it is largely flattened. Close the lid on the grill and cook for 15 minutes.
Carefully remove the brick. Using a spatula in one hand and a pair of tongs in the other, flip the chicken over. Close the lid and grill for another 15 minutes.
Flip the chicken over. Continue cooking, flipping the chicken every 10 minutes and closing the lid each time, until a thermometer inserted into the thickest part of one of the chicken’s thighs reads 165°F. The total cooking time should be 45-60 minutes.
Allow the chicken to rest for 10 minutes before carving and serving.