Preheat oven to 450 F.Beat eggs in a shallow dish (large enough to fit a sandwich). Set aside. Assemble sandwiches with ham and cheese. Spread the bread with strawberry jam. Slightly compress sandwich. Dip both sides of the sandwich in a mixture of egg, milk and vanilla, soaking for 5 seconds on each side. Put sandwiches on a greased tray, bake 6 minutes, flip and bake 5 more minutes, until browned and the cheese has melted. Sprinkle with powdered sugar before serving.