Preheat your oven to 180°C. Prepare a 25cm x 25cm slice tin by greasing it or lining it with parchment paper.
In a large bowl, whisk together the dry ingredients: wholemeal flour, white flour, baking soda, baking powder, and brown sugar.
In another bowl, beat the eggs, then mix in the oil, milk, mashed banana, and grated carrot until well combined.
Gradually add the wet mixture to the dry ingredients, folding gently until no dry streaks remain. Be careful not to overmix to keep the slice tender.
Pour the batter into your prepared tin, smoothing the surface with a spatula.
Bake for approximately 30 minutes, or until the top turns golden brown and a skewer inserted into the center comes out clean.
Remove from oven and allow to cool in the tin for 10 minutes, then transfer to a wire rack to cool completely.
Once cooled, cut into 5cm x 5cm squares. Store in an airtight container in the fridge for up to a week.