Preheat a grill to medium. Prepare the chicken: With your knife parallel to the cutting board, slice each chicken breast in half horizontally without cutting all the way through; open like a book. Cover with plastic wrap and pound until 1/2 inch thick.
Make the filling: Cook the bacon in a skillet until crisp; crumble. Mix 2 tablespoons bacon drippings with the melted butter in a bowl. Combine the buttermilk biscuit, bacon, half of the drippings mixture, the cheese, chives and parsley in a separate bowl. Stir in 2 to 3 tablespoons water; season with salt and pepper.
Season the pounded chicken on both sides with salt and pepper and brush with the remaining drippings mixture. Spoon the filling over the top, leaving a 1-inch border around the edges. Roll up the chicken toward the pointy end; tie with 3 pieces of kitchen twine.
Brush the grill grates with vegetable oil. Grill the chicken breasts, turning, until cooked through, 18 to 20 minutes, brushing with the barbecue sauce during the last 2 minutes of cooking. Transfer to a cutting board and let rest 5 minutes, then remove the twine and slice.