Using a flat meat tenderizer, gently pound the fatter part of the chicken out so that the breast is uniform thickness. Cut the chicken in half. Repeat with the other breast. Whisk together the buttermilk, egg, fish sauce, sriracha and ginger together and pour the mixture into a gallon ziplock bag with the chicken. Marinate for 2-8 hours. In a separate gallon ziplock bag, add the flour, potato starch, hoisin sauce, ginger, garlic and salt. Shake to mix the ingredients together well. After the chicken is done marinating, transfer the chicken pieces to the flour, allowing any excess buttermilk to drain off. Seal the bag and shake really well to coat all the chicken pieces in flour.
Heat at least 2-3″ of canola oil in a heavy bottomed pot or dutch oven to 360 degrees. Gently lower 2 pieces of chicken into the oil and fry for about 7 minutes, flipping once, until deeply golden brown and a probe thermometer registers 165 degrees. Transfer chicken to a wire rack set over a baking sheet. Repeat with the other two pieces. While the chicken is still warm, drizzle or brush the sauce liberally all over the chicken.
For the sauce
Combine all the ingredients in a nonstick skillet and simmer until thickened slightly to the consistency of a glaze. Set aside.
For the slaw:
Combine all the ingredients, cover, and chill until ready to use. Can be made 1 day in advance.
To assemble, lay slaw on the bottom bun of a toasted brioche burger bun. Top with chicken and pickles and top bun. Serve immediately.