At least 1 hour before grilling, soak planks in enough water to cover. Place weights on planks so they stay submerged during soaking.
In a small bowl combine brown sugar, paprika, mustard, salt, garlic powder, and cayenne pepper.
Spread spice mixture under the skins of chicken halves and over outsides of chicken halves; rub in with your fingers. Brush avocados with oil.
For a charcoal or gas grill, place planks on the rack of an uncovered grill directly over medium heat until planks begin to crackle and smoke. Meanwhile, place chicken halves, skin sides down, on the grill rack. Grill about 3 minutes or until grill marks appear. Place a chicken half, bone side down, on each plank. Lay bacon slices over chicken. Add avocado halves, cut sides up, to planks. Cover; grill for 30 minutes. Remove avocados from grill and set aside to cool. Cover; grill for 30 to 45 minutes more or until chicken is no longer pink (180 degrees F in a thigh muscle).
Transfer planks with chicken halves to two serving platters. Remove skins from avocado halves; slice avocados. Serve chicken with avocado slices and tomato slices.