Add chicken to a saucepan or a large deep skillet. Fill with enough water to cover the chicken and bring to a boil. Boil until the chicken is fully cooked (the internal temperature has reached 165 degrees). Drain any excess liquid from the pan and shred the chicken using two forks. Return the shredded chicken to the saucepan and mix with BBQ sauce. Cook on low heat until heated through.
Heat a skillet to medium heat. Spread a thin layer of butter on one side of each slice of bread. Place two of the slices into the skillet butter side down. Top each slice of bread in the skillet with 1 slice of cheese, pulled chicken, 2 dill pickle slices, and another slice of cheese. Cover with 2 of remaining bread slices, butter side up.
Cook the sandwiches over medium heat until browned on the bottom. Flip them over and cook until browned on the other side and the cheese has melted. Repeat the process with the 2 additional sandwiches.