Fragrant Dry Spice Rub
Combine all ingredients in small bowl.
Dissolve salt in 2 quarts cold water in large container. Immerse chicken in salted water and refrigerate until fully seasoned, about 1 hour. Remove chicken from brine and rinse inside and out with cool running water; pat dry with paper towels. Massage spice rub all over chicken, inside and out. Lift up skin over breast and rub spice rub directly onto meat.
While chicken is brining, soak 4 medium-sized wood chunks (we used 1/2 cup Mesquite chips) in water to cover for 1 hour; drain. When chicken is almost done brining, fill large chimney starter two-thirds full with charcoal (about 4 quarts, or 50 briquettes) and ignite. Burn until briquettes are covered with light gray ash, 15 to 20 minutes.
Empty coals into grill. Divide coals in half to form two piles on either side of grill; use long-handled tongs to move any stray coals into piles. Nestle 2 soaked wood chunks on top of each pile. Position grill rack over coals and cover grill. Heat rack for 5 minutes, then scrape clean with wire brush.
Open beer can and pour out (or drink) about 1/4 cup. If you don't buy beer in cans, wash a soda can and pour 9 oz. of beer into it. With church key can opener, punch two more large holes in top of can (for a total of three). Put can in Dutch oven* and slide chicken over can so that drumsticks reach down to bottom of can and chicken stands upright. Carry chicken to grill in Dutch oven and then wash so it can be used for cooked chicken.
Carefully lift chicken and can and place chicken and beer can on cool part of grill in a disposable aluminum pie pan, using ends of drumsticks to help steady bird - spread legs out so the legs and can form a tripod to balance chicken. Cover and grill-roast, - use clean cotton work gloves to lift chicken - rotating bird and can 180 degrees at halfway mark to ensure even cooking, until instant-read thermometer inserted into thickest part of thigh registers 170 to 175 degrees, 70 to 90 minutes.
Using gloves again, transfer chicken and can to Dutch oven*, making sure to keep can upright. Let rest 15 minutes; using gloves, carefully lift chicken off can and onto platter or cutting board. Discard remaining beer and can. Carve chicken and serve.
*It's easier to carry the chicken around in the Dutch oven - less chance of it sliding off a platter
Gas grill instructions:
Transfer the bird-on-a-can to your grill and place it in the center of the cooking grate, balancing the bird on its two legs and the can, like a tripod.
Grill chicken over Indirect High heat until the internal temperature registers 170°F in the breasts and 180°F in the thighs, or until the juices run clear, about 1-1/4 hours. Carefully remove from grill; do not spill contents of beer can, as it will be very hot. Let chicken rest for 10 minutes before lifting it from the beer can and carving it.