To make the batter: In a medium bowl, whisk together the flour, granulated sugar, baking powder, and salt. Add the eggs and milk and whisk until smooth. Set aside.
To make each sandwich: Sprinkle about 1/4 cup of the Gruyère over 1 slice of bread. Top with 2 slices ham and 2 slices turkey, then sprinkle 1/4 cup Gruyère over the meat. Place a second slice of bread on top.
To finish: Pour 3 inches of oil into a deep saucepan or Dutch oven. Have ready a wire rack lined with paper towels to use for draining. Heat the oil to 350°F on a deep-fry thermometer.
Working with 1 sandwich at a time, carefully dip it in the batter, letting the excess batter drip back into the bowl.
Slowly lower a battered sandwich into the oil and fry until golden brown on the bottom, about 4 minutes. Using a spatula, flip the sandwich over and fry until golden brown on the other side, about 4 minutes. Remove the sandwich to the paper towel-lined rack to absorb the excess grease. Repeat to fry the remaining sandwiches.
Using a sifter or fine-mesh sieve, gently tap 2 teaspoons of the powdered sugar over each sandwich.
In a small, microwave-safe bowl, stir together the raspberry preserves and 2 teaspoons of water. Microwave on high for 30 seconds, until loosened.
Serve each sandwich with about 2 tablespoons of raspberry preserves for dipping.