Preheat the grill, setting the burners to medium. Oil the grate.
In a large bowl, whisk together the eggs, milk, sugar and vanilla.
Dip each bread slice in the egg mixture and soak for several seconds. Set aside on a plate.
Grill the bread slices for 1 minute, then rotate them a quarter-turn and continue to grill for 1 minute. Flip the bread and top with the marshmallows and cookie pieces. Close the lid and cook for 1 to 2 minutes, checking regularly, until the French toast is cooked through and the chocolate is melted.
Serve topped with banana slices and drizzled with maple syrup.