Set up your grill for indirect heat by lighting burners on one side only and aiming for a temperature of 375 degrees Fahrenheit, or preheat your oven to 375 degrees. Place a drip pan filled with one cup of water under the unlit side of the grill or on the lower oven rack to catch drippings.

Combine brown sugar, paprika, garlic powder, onion powder, salt, black pepper, thyme, and cayenne in a small bowl, mixing thoroughly with a fork.

Remove the giblets from the chicken cavity if they are included, then pat the entire bird dry inside and out with paper towels.

Rub the olive oil all over the chicken, coating every surface including under the wings and around the legs.

Sprinkle the spice rub generously over the entire chicken, using your hands to massage it into the skin and ensure even coverage, including the bottom and inside the cavity.

Open the beer can and pour out about half, leaving roughly 6 ounces in the can. Use a can opener to punch two or three additional holes in the top of the can.

Hold the chicken upright with the cavity opening facing down, then lower it onto the beer can so the can slides into the cavity. The chicken should sit upright with the can supporting it and the legs spread out like a tripod. Tuck the wing tips behind the back to prevent them from burning.

Place the chicken on the unlit side of the grill or on a rimmed baking sheet in the oven, making sure it stands upright and stable. Close the grill lid or oven door and cook for 60 to 75 minutes without opening it. The chicken is done when a meat thermometer inserted into the thickest part of the thigh reads 165 degrees Fahrenheit.

Carefully remove the chicken from the grill or oven using tongs to grasp the can. Transfer the chicken, still on the can, to a cutting board and let it rest for 10 minutes. Remove the beer can by holding the chicken steady with one hand and pulling the can out with tongs, then carve and serve immediately.