Step 1

Adjust oven rack to upper-middle position and preheat oven to 450 degrees.

Step 2

Whisk together eggs, cream, sugar, salt, dry mustard, and cayenne in shallow dish until combined. Stir jam and Dijon mustard together in a small bowl.

Step 3

Spread 1 teaspoon jam mixture on 1 side of each slice of toast. Layer slices of cheese, ham, and turkey over jam on 6 slices of toast.

Step 4

Repeat with the second layer of cheese, ham, and turkey. Add final layer of cheese and top with remaining toast, with jam side facing cheese. Using hands, lightly press down on sandwiches.

Step 5

Pour oil into rimmed baking sheet, place in oven, and heat until just smoking, about 7 minutes.  Meanwhile, using 2 hands, coat each sandwich with egg mixture and transfer to large plate.

Step 6

Transfer sandwiches to preheated baking sheet and bake until golden brown on both sides, 4 to 5 minutes per side, using spatula to flip. Remove from oven, sprinkle with powdered sugar, and serve immediately with remaining jam mixture.