In a bowl, stir together eggs, milk, vanilla, and sugar until well blended and sugar is dissolved. In a separate bowl, mix flour with baking powder and salt. Gradually stir flour mixture into egg mixture until no lumps remain. Set aside.
Heat oil in deep-fryer to 375 degrees F (190 degrees C).
Make sandwiches with the bread, spreading jam evenly in the center. Trim away crusts, and cut each sandwich into quarters, wiping away any excess jam. Dip each quarter into batter, coating completely.
Working in batches to avoid overcrowding, fry sandwiches for 1 to 3 minutes, or until puffy and golden brown. Drain and cool on a paper towel-lined wire rack. Sprinkle with powdered sugar. Serve warm.