Preheat the oven to 190°C (170°C fan), gas mark 5. Line two baking trays with baking parchment or silicone baking mats.
Cream the butter, caster sugar and light brown sugar together in a large bowl for 2–3 minutes, until pale and fluffy.
Beat in the egg and vanilla extract until just combined.
Stir in the mashed banana until evenly mixed but still slightly lumpy.
In a separate bowl, whisk together the plain flour, bicarbonate of soda and salt.
Fold the dry ingredients into the banana mixture until just combined, taking care not to overmix.
Drop rounded tablespoons of dough about 5 cm apart onto the prepared trays.
Bake for 10–12 minutes, until the edges are lightly golden and the centres spring back when pressed.
Allow the biscuits to cool on the trays for 3–5 minutes, then transfer to a wire rack to cool completely.