Cut off the top of the can and pour in 12 ounces of beer.
Add the bay leaves and thyme. Set aside.
Make the rub by combining 2 tablespoons brown sugar with the paprika, salt, and pepper.
Make the turkey baste. In a separate bowl, combine the remaining 2 tablespoons brown sugar , the ketchup, vinegar, 2 tablespoons of beer, hot sauce, and cayenne.
Prepare smoker for a 6-hour smoke at 250 F.
Rub surface of the turkey with the spice rub . Try to get as much as you can under the skin, particularly over the breast meat.
Stand beer can up on the smoker grate and gently set the turkey over the can so that the can is completely inside the cavity of the turkey. Make sure that the turkey stands easily and doesn't sway. The turkey needs to be stable.
Once you have the turkey in the smoker, let it cook for about 6 hours at a temperature around 250 F. Baste the turkey with the basting mixture every 2 hours during the cooking process. Check for an internal temperature in the thigh of about 165 F/74 C. This is when the bird is done.
When cooked, remove from heat and let sit for a few minutes before carving.