Preheat the oven to 350°F. Cut the day-old bread into 1-inch cubes and place them into a small bowl. Pour the melted butter over the bread cubes.
In another bowl, whisk together the eggs, milk, sugar, cinnamon, nutmeg, and vanilla. Pour the mixture over the bread cubes and let them soak for at least 10 minutes.
Grease a 9x9-inch pan with butter or cooking spray. Pour the soaked bread cubes and the remaining egg mixture into the pan, spreading them into an even layer. Add the chopped bananas last, pressing them into the open spaces between the bread cubes.
Bake the bread pudding for 30-40 minutes, or until a toothpick inserted into the center comes out clean.
While the bread pudding is baking, in a small skillet over medium-high heat, stir together the butter, sugar, cornstarch, milk, and corn syrup. Bring the mixture to a rolling boil, stirring occasionally.
Stir the vanilla into the milk glaze and remove it from the heat. Allow the glaze to cool until warm and then pour it over the baked banana bread pudding. Serve warm or at room temperature; the glaze will stiffen and set once cooled. Enjoy!