Bring a large pot of water to a boil over high heat. When the water reaches a rolling boil, gently add chicken breasts, cover pot with a fitted lid and remove from heat. Allow chicken to cook for approximately 30 minutes. Carefully remove chicken from the pot to a plate and allow it to cool before shredding with a fork (or fingers). Meanwhile, while the chicken is cooking, prepare the BBQ sauce. In a medium size sauce pan heat butter over medium high heat. Add grated onion and minced garlic and mix well. Cook onion and garlic over medium heat, stirring frequently, until onions are soft, approximately 4-5 minutes. Gently mix in remaining ingredients and bring sauce to a simmer. Allow mixture to cook on low for 15-20 minutes, taste, and make any additions you see necessary (sugar, vinegar, salt). Remove from heat and carefully stir in shredded chicken. Set aside.
For the pickled red onion:
Placed sliced red onion in a medium bowl. In a small pot bring vinegar, water, sugar and salt to a low boil. Carefully pour over the sliced red onion and allow onions to sit in the vinegar mixture for at least 30 minutes to 1 hour. When ready to eat, drain liquid and rinse with cool water.
For the coleslaw:
In a large mixing bowl combine all ingredients for the coleslaw and mix well. Taste and add additional salt and pepper if necessary.
Build the sandwiches:
Lightly toast the brioche buns and pile on the toppings. First the chicken, then the onions and finally, the coleslaw.