Heat oven to 350 degrees F. Spread out angel cake cubes on a large rimmed baking sheet. Bake 20 minutes, tossing three times, until lightly toasted and dry. Turn off oven.
Whisk milk, eggs, sugar, rum and vanilla extract in a large bowl until sugar is dissolved. Gently stir in cake cubes and banana slices; pour into an 8-inch square baking dish. Cover dish with foil. Let stand 30 minutes at room temperature at room temperature for cake cubes to absorb most of the milk mixture.
Heat oven to 350 degrees F. Bake, covered, 30 minutes. Remove foil; continue to bake 25 minutes or until pudding is set.
Let stand 15 minutes. Dust top with confectioners sugar. Server with small scoops of vanilla ice cream.