Prepare the Ritz Cracker Coating: In a food processor, pulse 50 Ritz crackers until you achieve coarse crumbs. Transfer the crumbs to a shallow dish and set aside.
Make the Custard Mixture: In a mixing bowl, whisk together 2/3 cup of milk, 2 large eggs, and 1/2 cup of mayonnaise until smooth and well combined.
Assemble the Sandwiches: Take 4 slices of hearty white sandwich bread and layer 1 slice of American cheese on each. Top with the remaining 4 slices of bread to create 4 sandwiches.
Dip the Sandwiches: Carefully dip each sandwich into the custard mixture, ensuring both sides are well coated.
Coat with Ritz Cracker Crumbs: Dredge each custard-coated sandwich in the Ritz cracker crumbs, pressing gently to adhere. Make sure each sandwich is evenly coated for that perfect crunch.
Heat the Oil: In a frying pan, heat a generous amount of vegetable oil over medium heat. You want enough oil to cover the bottom of the pan.
Fry the Frenchies: Once the oil is hot, carefully place the coated sandwiches in the pan. Fry until golden brown, about 3-4 minutes per side. Flip with a spatula to ensure even cooking.
Drain and Serve: Once cooked, transfer the Cheese Frenchies to a plate lined with paper towels to absorb excess oil. Cut each sandwich into triangles or quarters for easy serving.