Preparation

• Make the chicken: One at a time, place the chicken breasts in a resealable freezer bag; using a meat mallet or rolling pin, pound them to an even ¼-inch thickness. Cut each pounded breast in half. (If the pieces seem too large to fit in a bun, trim off the pointy ends; you can cook those as chicken nuggets at the same time you're cooking the breasts.)
• Pour the cold water into a medium bowl; add the 2 tablespoons of the salt and the sugar and whisk until dissolved. Place the chicken breasts in the brine, cover the bowl, and refrigerate for at least 4 hours or overnight (the longer you brine, the saltier the chicken will be).
• Line a plate with a few layers of paper towels and set it next to the stove. Pour the oil into a large, high-sided pot or Dutch oven until it reaches a depth of about 1 inch. Heat the oil over medium-high heat until shimmering (about 350°F). (If the end of a wooden spoon or cube of bread sizzles vigorously when you dip it in, the oil is hot enough.)
• In a medium bowl, whisk the egg and milk until evenly combined.
• In another medium bowl, whisk the flour, cornstarch, baking powder, garlic powder, pepper, Accent, paprika, and the remaining ¾ teaspoon salt. Drizzle 4 tablespoons of the egg mixture into the flour mixture and mix with a fork until the flour mixture is evenly clumpy.
• Line a baking sheet with aluminum foil for easy cleanup. Remove the chicken from the brine and pat dry with paper towels. One at a time, dip each breast into the egg mixture, letting any excess drip off, and then into the flour mixture, turning a few times and pressing firmly so that the clumps adhere. Set the breaded chicken on the lined baking sheet and repeat with the remaining breasts.
• Carefully place a few pieces of breaded chicken into the hot oil without crowding the pot. Cook, turning a few times, until golden brown and crispy on both sides, 4 to 5 minutes total. Place the cooked chicken on the paper towel-lined plate to drain. Fry the remaining breasts in the same manner, adjusting the heat as necessary to maintain the temperature of the oil.
• Make the sauce: In a small bowl, whisk together the mayonnaise, barbecue sauce, and mustard.
• Assemble the sandwiches: Spread some of the sauce on each top and bottom bun. Top each bottom bun with a piece of crispy chicken and a few pickles. Close the sandwiches and serve immediately.