In a frying pan set over a low heat, melt the butter, add the brown sugar, and give it a stir.
Leave it to simmer for 2 minutes.
To make the batter, crack the egg in a mixing bowl, and use a whisk to beat it well.
Add the caster sugar to the egg, and beat well again to get a pale yellow colour.
Pour in the milk and vanilla extract, and mix.
Sift in the flour, and use a spatula to combine all the ingredients well to get a thin batter.
Peel and slice the bananas in 3 slices vertically.
Arrange the banana slices in the pan over the caramel, then pour over the batter, spreading it evenly.
Place a lid on the pan, and leave it to cook for 15 minutes until bubbles start forming on the top like with an American pancake.
Brush a large plate with a bit of melted butter, then remove the lid, and place the plate over the pan.
Quickly turn the pan so that the bottom of the cake in on the plate.
Carefully slide the cake back in the pan, so that the bottom of the cake can brown lightly too - leave it too cook for a further 2 minutes.
Remove from the heat, and leave the cake to cool down completely.