1Set up your grill for indirect cooking at about 350 to 375°F: light one side only on a gas grill, or bank the coals to one side on a charcoal grill. Put a foil pan on the unlit side to catch drips if you like.
2Stir the brown sugar, smoked paprika, garlic powder, onion powder, kosher salt, black pepper, cayenne and thyme together in a small bowl.
3Pat the chicken completely dry with paper towels, then rub it all over with the vegetable oil. Sprinkle the spice rub over the outside and inside of the bird, working some under the skin of the breasts.
4Open the beer and drink or pour off about half of it, leaving roughly 6 oz in the can. Drop the smashed garlic cloves into the can.
5Lower the chicken cavity down onto the can so the bird sits upright, with the can fully inside the cavity. Pull the legs forward so the chicken balances on its two legs and the can like a tripod.
6Carefully stand the chicken on the unlit side of the grill, over the drip pan if using. Close the lid and cook for 75 to 90 minutes, keeping the grill near 350°F, until the skin is deep golden brown.
7Check the temperature: the thickest part of the breast should read 165°F and the thigh 175°F. If the skin darkens too fast, tent the breast loosely with foil.
8Using tongs and a sturdy spatula or heatproof gloves, lift the chicken on its can onto a platter, keeping it upright so no hot beer spills. Rest it for 15 minutes.
9Lift the chicken off the can with tongs while holding the can steady with a towel; the can will be very hot and the liquid inside scalding. Discard the can and its contents, carve, and serve.