Preheat the Oven: Set your oven to 350°F (180°C). Grease a 4.5 x 8.5-inch loaf tin and line it with parchment paper, leaving overhang for easy removal.
Mix the Wet Ingredients: In a large bowl, whisk together the light brown soft sugar, caster sugar, oil, non-dairy milk, vinegar, vanilla extract, and lemon (or orange) zest until smooth.
Prepare the Dry Ingredients: In a separate bowl, sift the plain flour, wholemeal flour, baking powder, bicarbonate of soda, cinnamon, ginger, nutmeg, and salt.
Combine Wet and Dry Ingredients: Slowly add the dry ingredients into the wet ingredients and whisk gently until just combined. Avoid over-mixing to ensure a light loaf.
Add Grated Carrots: Fold the grated carrots into the batter until evenly mixed. Optionally, fold in raisins or chopped walnuts for added texture.
Bake the Cake: Pour the batter into the prepared loaf tin and spread it level. Bake for about 1 hour, or until a skewer inserted into the center comes out clean.
Cool the Cake: Let the cake cool in the tin for 20 minutes, then carefully lift it out using the parchment paper and place it on a wire rack to cool completely.
Prepare the Lemon Glaze: While the cake cools, whisk the sifted icing sugar, non-dairy yogurt (optional), and lemon juice in a small bowl until smooth. Adjust with extra lemon juice if necessary to achieve a thick but pourable consistency.
Glaze the Cake: Once the cake is completely cooled, pour the glaze over the top and spread it evenly. Serve and enjoy!