Rinse the chicken fillet, dry with a paper towel, and sprinkle with salt and pepper on all sides.
Heat butter in a large skillet over medium-high heat. Place the fillets in a preheated pan and fry for about 2 minutes on each side, until golden brown. Add barbecue sauce, 1/2 cup vinegar, and hot sauce to a skillet, bring to a boil. Reduce heat to medium-low, cover skillet, and cook fillets for about 10 minutes. Transfer the finished meat to a plate and cover with foil. Continue to cook the sauce in the pan for another 3 minutes. Cover the prepared sauce and set aside.
In a separate bowl, combine mayonnaise, buttermilk (sour milk), and 1 tablespoon of vinegar. Put cabbage in this dressing, season with salt and pepper to taste, mix well.
Tear the chicken meat into small pieces and put these pieces in a pan with hot sauce, stir.
Cut the buns in half and place the fillet with sauce and cabbage between the two halves. Serve the chicken burgers right away.