Line large rimmed baking sheet with parchment paper. Spread cracker crumbs in shallow dish. Whisk milk and eggs together in medium bowl.
Spread 1 Tbsp. of the mayonnaise on one side of each slice of bread. Arrange 1 1/2 slices of cheese on 4 of the slices of bread and top with remaining 4 slices bread, with the mayonnaise side facing the cheese. Cut each sandwich diagonally into quarters.
One at a time, dip sandwich quarters into egg mixture, then coat with the cracker crumbs, pressing to adhere. Put on the prepared baking sheet and refrigerate til set, about 1 hour.
Pour oil into large Dutch oven til it measures 2 inches deep. Heat oil to 375 degrees over medium high heat.
Lay half of the chilled sandwich quarters in oil and fry til golden brown on both sides, 2 to 3 minutes, flipping halfway through. Transfer sandwich quarters to a large paper towel lined plate to drain briefly. Repeat with remaining sandwich quarters. Serve. Serves 4