Adjust oven rack to the middle position and preheat oven to 200 degrees. Line a rimmed baking sheet with foil for easy cleanup. Coat a wire rack with nonstick spray and set on baking sheet.
Cover a cutting board completely with a piece of plastic wrap. Cut pork into 4 pieces, aiming for them to be similar in weight if not shape (for example, cut tapered tail pieces slightly thicker than middle medallions).
Arrange all four pieces in a single layer, cut side up, on the plastic wrap-covered cutting board. Cover with a second piece of plastic wrap. Pound pork to an even thickness, about 1/4-inch. Remove top layer of plastic wrap and season pork with salt and pepper.
In a food processor, pulse bread and saltines until finely ground, about 12 pulses. Transfer to shallow dish and season with salt and pepper (I like 1/4 teaspoon salt and 1/8 teaspoon pepper).
In a second shallow dish, spread flour in second shallow dish and season with salt and pepper (I like 1/4 teaspoon salt and 1/8 teaspoon pepper).
In a third shallow dish, whisk eggs and mayonnaise together. Arrange the shallow dishes in this order: Flour, egg mixture, then crumbs.
Working with 1 cutlet a time, dredge cutlets in flour, dip in egg mixture, then coat with crumbs, pressing gently to adhere. Return cutlets to cutting board or plate and let dry for 5 minutes.
Heat 1/2 cup oil in 12-inch nonstick skillet over medium heat until shimmering. Working with 2 cutlets at a time, fry cutlets until deep golden brown and crisp, 2 to 3 minutes per side.
Transfer fried cutlets to wire rack on prepared baking sheet and keep warm in oven. Discard oil, wipe out skillet, and repeat with remaining 1/2 cup oil and remaining 2 cutlets. Serve on buns with lettuce, tomato, and mayonnaise.