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.
Cut tenderloin into 4 pieces (cut tapered tail piece slightly thicker than middle medallions). Lay pork medallions flat, cut side up, on cutting board.
Cover with plastic wrap and pound and pound to 1/4-inch thickness with meat pounder. Pat cutlets dry with paper towels and sprinkle 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).
Spread flour in second shallow dish and season with salt and pepper (I like 1/4 teaspoon salt and 1/8 teaspoon pepper).
Whisk eggs, mayonnaise, and salt and pepper together in third shallow dish (I like 1/4 teaspoon salt and 1/8 teaspoon pepper). Arrange the shallow dishes in this order: Flour, egg and mayonnaise mixture, then crumbs.
Working with 1 cutlet a time, dredge cutlets in flour, dip in eggs 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 extra mayonnaise.