In a large bowl, combine the ingredients for the custard. Whisk until frothy. Add bread to the mixture, press it gently and let it soak for at least 10 minutes. Preheat oven to 375 degrees. Grease a 2-quart baking dish with butter. Line the bottom of the dish with a layer of banana coins, then spoon about ½ of the soaked bread over them. Top with banana coins and spoon the remaining soaked bread over the top. Bake for 50-60 minutes. While the bread pudding is in the oven, make the sauce. Melt the butter in a small saucepan set over medium heat. Add the ingredients for the sauce and whisk until combined. Lower the heat until it simmers; keep whisking, until a thick sauce has developed and coats the back of a spoon, approximately 5 to 8 minutes. Remove from heat and keep warm. Allow pudding to cool down to warm before serving. Drizzle the sauce over the top with some caramelized bananas and vanilla ice cream.