Season the chicken with salt and black pepper and refrigerate uncovered for 30 minutes.
Place the flour and crushed saltines in separate shallow bowls, then beat the eggs with 1 tablespoon Dijon in a third bowl.
Dredge each cutlet in flour, dip it in the mustard egg and press it into the saltines, then rest the coated chicken for 8 minutes.
Stir the mayonnaise, honey and remaining Dijon until smooth.
Heat the peanut oil in a wide heavy skillet to 345°F and fry 2 cutlets at a time for 3 to 4 minutes per side, maintaining 330°F to 345°F, until crisp and 165°F inside.
Drain the chicken on a wire rack and rest it for 4 minutes.
Toast the potato rolls and assemble each with honey-mustard mayo, 2 lettuce leaves and 1 saltine-crusted cutlet.