Preheat oven to 350° F. To caramelize the apple banana, peel and cut into ½" cubes and toss with the brown sugar. Heat a non-stick skillet on medium heat and melt two tablespoons of butter, then add the coated apple banana. Stir occasionally and continue cooking until the banana is golden brown. Remove and set aside.
Traditionally, bread pudding is made with day old bread, but in Hawai‘i bread tends to stay moist – you can put your sweetbread slices in the oven on a cookie sheet for 5 to 7 minutes to dry it out a little while the apple banana is browning (this is optional). Tear the bread slices into chunks, and put into a 8" square baking pan either greased with butter or sprayed with cooking spray. Sprinkle the broken sweetbread with the caramelized apple banana and drizzle the melted two tablespoons of butter over the top.
In a bowl, mix the eggs, milk, coconut milk, sugar, cinnamon and vanilla extract together – then pour over the sweetbread. Make sure every piece of the sweetbread soaks up some of the liquid. If not, rotate the pan to distribute the liquid evenly, and press down lightly with a spatula. Bake 40 to 45 minutes.