Line a rimmed baking sheet with parchment paper and set aside. Place the pulsed Ritz cracker crumbs in a shallow dish.

In a medium bowl, whisk together the eggs and the 2/3 cup milk until blended.

Spread 1 tablespoon mayonnaise on one side of each of the 8 slices of bread.

On 4 of the bread slices (mayonnaise side up), place 1 1/2 slices of American cheese on each. Top each with one of the remaining 4 bread slices, mayonnaise side down, to form 4 sandwiches.

Cut each sandwich diagonally into quarters (you will have 16 pieces).

Working one piece at a time, dip a sandwich quarter into the egg-and-milk mixture, turning to coat all sides, then press it into the Ritz cracker crumbs to coat completely. Place coated pieces on the prepared parchment-lined baking sheet. Repeat with remaining pieces.

Refrigerate the coated sandwich pieces on the baking sheet for 1 hour.

Pour about 2 inches of vegetable oil into a Dutch oven and heat the oil to 375°F.

Working in batches to avoid overcrowding, carefully fry the sandwich quarters for 2 to 3 minutes each, flipping once about halfway through, until golden brown.

Transfer fried pieces to a paper towel–lined plate to drain before serving.