Cook the prosciutto and sage: Prepare a baking sheet fitted with a kitchen towel. In a large skillet, heat about 1/2 cup of the oil over low heat. When the oil begins to smoke lightly, add the prosciutto slices to the oil in the pan and cook until crispy, 1 to 2 minutes. Use a slotted spoon or spatula to remove them. Place them on the kitchen towel to drain. Next, add the sage leaves and cook until they pale in color and become crispy, 30 seconds to 1 minute. Remove and season them with salt. Set aside.

Make the sauce: In a small bowl, whisk together the mayonnaise and red wine vinegar.

Prepare the chicken cutlets: Prepare another baking sheet fitted with parchment paper. Sprinkle both sides of the chicken cutlets with the oregano and salt. Put the eggs in one bowl and the breadcrumbs in another. Dip each piece of chicken in the egg (on both sides) and then the breadcrumbs. Shake any excess off each one.

Cook the chicken cutlets: Add the remaining oil to the pan used to fry the prosciutto and heat it until it smokes lightly. Use a pair of tongs to carefully add a couple of the chicken cutlets to the oil in a single layer. Cook over medium heat on the first side until golden brown, 3 to 5 minutes. Turn on their second side and cook 3 to 5 additional minutes. Cut a small incision in the middle of each cutlet to make sure they are completely cooked. If any are underdone, place in a warm oven to cook for a few additional minutes. Repeat the process with the remaining cutlets.

Assemble the sandwiches: Spoon a generous helping of the mayonnaise mixture on the bottom half of the roll. Top with a mound of lettuce. Top with the chicken cutlets, some Parmesan, sage leaves and prosciutto slices. Drizzle with additional mayonnaise if desired. Top with the other half of the roll.