Place the chicken with 3/4 cup pickle brine in a covered dish and refrigerate for 1 to 4 hours.
Whisk the cornmeal, flour, paprika, salt and black pepper in one shallow bowl and beat the eggs with 2 tablespoons pickle brine in another.
Drain the chicken, dip each cutlet in egg and press it into the cornmeal mixture until fully coated, then rest the pieces for 8 minutes.
Stir the cabbage, mayonnaise and remaining 2 tablespoons pickle brine together and refrigerate the slaw while the chicken cooks.
Heat the vegetable oil in a deep heavy pot to 350°F and fry 2 cutlets at a time for 4 to 6 minutes, turning once, until the coating is crisp and the chicken reaches 165°F.
Drain the cutlets on a wire rack and wait 4 minutes so trapped steam does not soften the crust.
Toast the buns cut-side down for 1 minute and build each sandwich with 1 cutlet and 1/2 cup pickle slaw.