Place the buttermilk in a large, shallow dish; set aside. Combine the flour, sugar, measured salt, paprika, measured pepper, baking powder, and ground mustard in another large, shallow dish and whisk to combine; set aside.

Cut each chicken breast in half horizontally. Place 1 piece between 2 sheets of plastic wrap or in a resealable plastic bag and pound to an even thickness (about 1/4 to 1/2 inch thick) with a meat pounder or the bottom of a pan; set aside. Repeat with the remaining pieces of chicken. Season the chicken generously on both sides with salt and pepper.

To bread the chicken, dip 1 piece in the buttermilk and let the excess drip off. Dredge in the flour mixture and shake off any excess. Set the coated chicken on a baking sheet and repeat with the remaining pieces.

Heat the oil in a large frying pan over medium-high heat until shimmering but not smoking (about 350°F on a deep-frying/candy thermometer). Meanwhile, fit a wire rack over a second baking sheet; set aside.

When the oil is ready, add 2 pieces of the chicken and fry until golden brown and cooked through, about 2 minutes per side. Remove to the wire rack and season with salt. Repeat with the remaining chicken. Cover the fried chicken loosely with a piece of aluminum foil; set aside.

To serve, stem and thinly slice the pickled peppers. Evenly divide the mayonnaise among the rolls, spreading it all over both of the cut sides. Place the chicken pieces on the rolls, top with the lettuce and peppers, and serve immediately.