First, preheat the oven to 450 degrees. In a small bowl, mix together the Dijon mustard and strawberry jam together and set aside. Next, make the custard. Whisk together eggs, dry mustard, sugar, salt, and cayenne pepper in a shallow dish and set aside. To assemble the sandwich, spread the jam/mustard mixture over one slice of bread. Then top with cheese, ham, and turkey in that order. If you have enough of everything, repeat the layer of cheese, ham, and turkey, and finally, finish with one more layer of cheese. Spread more jam/mustard on the second piece of toasted bread, and top the sandwich with it, jam side down, on top of the cheese. Cheese on both sides of the bread will glue the sandwich together as it bakes. Give the sandwich a gentle but firm press with the palm of your hand. For oven-frying the sandwich: Pour some vegetable oil in a rimmed sheet pan and put it in the oven on the upper-middle rack. Let the oil heat up until almost smoking, about 7 minutes. To coat the Monte Cristo in batter, carefully dip both sides of the sandwich in the custard. When ready to bake, carefully move them to the pre-heated, oiled baking tray. Bake the sandwiches for 4 to 6 minutes on each side, using a spatula to flip them as they brown. Finally, remove them from the oven and serve immediately, dusted with powdered sugar and extra jam on the side.