Preheat the oven 350°F (180°C). Sift the flour and baking powder into a bowl. Add the sugar. Mix together the beaten egg, mashed banana, milk, vanilla extract and melted butter. Pour mixture into the dry ingredients and stir until combined. Pour this wet dough into a 9-inch pie dish and place the dish on a baking sheet.
To make the topping, put the brown sugar, corn syrup and boiling water in a saucepan. Bring to the boil and then drizzle over the pudding. Bake for 30–40 minutes, or until it feels slightly firm in the center. Serve with softly whipped cream or vanilla ice cream. If you’re not going to serve the pudding immediately, keep it somewhere warm until you are ready—it sits quite happily.