Place the chicken breasts in a resealable freezer bag. Pound them to an even ¼-inch thickness and cut each breast in half. Trim off pointy ends as necessary.

Pour the cold water into a medium bowl. Add 2 tablespoons of salt and sugar, whisk until well dissolved. Immerse the chicken breasts in this brine, cover, and refrigerate for at least 4 hours or overnight.

Place paper towels on a plate to catch excess oil. Pour oil into a pot or Dutch oven, heating to about 350°F (175°C). Test if it's hot enough by dipping a wooden spoon into the oil.

In a medium bowl, whisk together egg and milk. In another medium bowl, combine flour, cornstarch, baking powder, garlic powder, black pepper, Accent, paprika, and ¾ teaspoon of salt. Drizzle 4 tablespoons of the egg mixture into the flour mixture and mix until clumpy.

Line a baking sheet with aluminum foil. Remove chicken from the brine and pat dry. Dip each chicken breast in the egg mixture, then coat with the flour mixture. Place breaded pieces on the lined baking sheet.

Carefully place breaded chicken into the hot oil, fry until golden brown (about 4 to 5 minutes). Transfer cooked chicken to the paper towel-lined plate.

Whisk together mayonnaise, barbecue sauce, and Dijon mustard in a small bowl until well blended.

Spread sauce on both halves of each bun. Place a chicken piece on each bottom bun, add pickles, and cap with the top bun. Serve immediately.