Preheat the oven to 180°C/350°F/gas mark 4. Grease an approx 11.5 x 21.5 cm / 4.5 x 8.5 in loaf tin and line it with a strip of baking parchment that comes a little way above the sides so that you can easily lift the cake out once it is baked.
Grate the carrot using the coarse side of the grater. Squeeze the grated carrot dry using paper towels; there should be about 120g left after squeezing. Set aside.
Mash the bananas in a large bowl then whisk in the oil, milk, sugar, vanilla and vinegar until well combined.
Sift together the plain and wholemeal flours, baking powder, bicarbonate of soda, salt, cinnamon, ginger and nutmeg.
Tip the dry ingredients into the wet and stir with a balloon whisk until no dry lumps remain. Don't over-beat the mixture.
Add the grated carrot and stir through.
Pour the batter into the prepared tin and spread level. Bake for about 60-70 minutes until a skewer inserted into the centre comes out clean. Cover the cake loosely with tin foil part way through baking if the top starts to get too dark.
Leave to cool in the tin for 20 minutes then carefully turn out onto a wire rack and leave to cool completely.