Combine eggs, milk and vanilla extract and beat extremely well. Heat approximately 1/2 in of oil to medium high heat. Dip only one side of two slices of bread in egg mixture, lay in hot oil, dipped side down. Quickly put a slice of Swiss cheese on each bread, then ham, then another slice of cheese. Dip only one side of 2 more slices of bread in egg mixture and put dry side down on top of cheese.
Lift carefully with spatula checking to see if bread is evenly browned. If toasted slightly dark brown then very! very! carefully turn over each sandwich and toast that side.
Put finished sandwiches on paper towel to drain excess oil off. Repeat process till all are done being kept warm in oven. Put each sandwich on individual serving plates and spread a thin amount of blackberry jam on top. Sprinkle with a little white powdered sugar and serve.