In a large shallow bowl, using a fork, beat the eggs until lightly frothy. Stir in the milk and set aside. Peel the bananas into a small bowl and mash them with a fork. Stir in the walnuts and the nutmeg.
Spread the banana mixture over the 1/2 of the bread slices, leaving a 1/4-inch border uncovered on all edges. Top with the remaining bread slices and press down gently to seal.
Place 2 sandwiches in the egg mixture and press down gently. Turn over and let soak for a moment until evenly saturated on both sides. Repeat with the other 2 remaining sandwiches.
In a frying pan or on a griddle, melt 2 tablespoons of butter over medium heat. Place the sandwiches on griddle and fry until the undersides are golden brown, about 2 minutes. Add the remaining butter, flip the sandwiches with a spatula and fry until golden brown.
To serve place the toast on plates and dust with confectioners' sugar. Serve with maple syrup or jam.
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