Place each chicken cutlet between plastic wrap and pound to 1/4 inch thick with a meat mallet.
Set up a dredging station with three shallow bowls: one with flour, one with beaten eggs, and one with a mixture of breadcrumbs, Parmesan, salt, and pepper.
Dredge each cutlet in flour, shaking off excess, then dip in egg, and coat thoroughly with breadcrumb mixture.
Heat oil in a large skillet over medium heat. Fry cutlets in batches until golden brown and cooked through, about 3-4 minutes per side.
Mix mayonnaise and Dijon mustard in a small bowl. Spread the mixture on each slice of bread.
Assemble the sandwich: add a layer of lettuce, a few slices of tomato, and a chicken cutlet to each sandwich.
Top with the remaining bread slices, cut in half, and serve.
Notes
To add a spicy kick, consider adding some hot sauce to the mayonnaise mixture.