Preheat the oven to 325 ºF (160 ºC) and grease a 9-x-5-inch (2L) loaf pan.
Whisk the banana, melted butter, granulated and brown sugar together until well-blended. Whisk in the egg and vanilla.
In a separate bowl, sift the flour, baking soda and salt. Add this to the banana mixture and stir just until blended. Stir in the chocolate chips or walnut pieces (if using). Scrape the batter into the prepared loaf tin and sprinkle the top with a little sugar. Bake for about 75 minutes, until a tester inserted in the centre of the cake comes out clean. Cool the cake for 20 minutes in the tin, then turn it out to cool completely on a cooling rack.