Preheat oven to 350°F. Lightly grease an 8x5" loaf tin.
In a large mixing bowl, stir together white sugar, brown sugar, plant milk, oil, and vanilla extract. Fold in the grated carrots.
In a separate bowl, whisk together dry ingredients: all-purpose flour, baking soda, baking powder, salt, cinnamon, ginger, nutmeg, and allspice. Add dry ingredients into the wet mixture and stir to combine.
Optionally, fold in chopped nuts. Pour the batter into the prepared loaf tin and smooth off the top.
Bake for 50-55 minutes, until the loaf is fully set and starting to pull away slightly from the sides of the loaf tin. Temperature should register between 200-205°F to ensure it's fully cooked.
Allow the loaf to cool for about 10 minutes in the tin, then transfer it to a wire cooling rack and allow it to cool fully before icing.
Whisk together all ingredients for the icing. Drizzle the icing evenly over the top of the loaf. It will take about 30-60 minutes to set fully.