Preheat your oven to 350°F (180°C) or gas mark 4. Grease a 4.5 x 8.5-inch (11.5 x 21.5 cm) loaf tin and line it with a strip of baking parchment, leaving an overhang on the sides for easy removal.
In a large bowl, whisk together the light brown sugar, caster sugar, oil, milk, vinegar, vanilla extract, and your choice of lemon or orange zest.
In a separate bowl, sift the plain flour, wholemeal flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt.
Gently whisk the dry ingredients into the wet mixture until just combined and no flour pockets remain. Avoid over-mixing, as it can result in a dense cake.
Fold the grated carrots into the batter until well distributed.
Pour the batter into the prepared loaf tin, smooth the top, and bake for around 1 hour, or until a skewer inserted into the center comes out clean.
Let the cake cool in the tin for 20 minutes before lifting it out using the parchment and transferring it to a wire rack to cool completely.