Place the chicken breasts in a skillet and add enough water to cover them halfway. Bring to a simmer over medium heat and cook for 12 to 15 minutes until the chicken reaches 165°F and is no longer pink in the center.
Remove the chicken from the skillet and let it cool slightly. Shred the meat using two forks.
Return the shredded chicken to the skillet. Pour in the barbecue sauce and stir to coat evenly. Heat over low heat for 3 to 4 minutes until warmed through.
In a mixing bowl, combine the shredded cabbage and grated carrots. Add the mayonnaise, apple cider vinegar, sugar, salt, and black pepper. Stir until the vegetables are evenly coated.
Split the brioche buns and divide the pulled chicken among the bottoms. Top each sandwich with a generous scoop of creamy slaw before adding the bun tops.