Preheat the oven to 350°F (180°C). Peel and finely grate the carrots.
Whisk together the flour, brown sugar, cane sugar, baking powder, baking soda, salt, cinnamon, and nutmeg in a large bowl.
Make a well in the center of the bowl. Add the applesauce, plant milk, oil, and vanilla to it. Stir the ingredients together with a spoon until they are mostly combined.
Gently fold in the carrots and walnuts with just a few turns to prevent over-mixing the batter.
Place 12 liners into a cupcake pan and then divide the batter evenly across the liners. Each one should be about ¾ full.
Bake the cupcakes in the oven for 20-25 minutes. They are done when a toothpick inserted into the middle of one of them comes out clean. You can either enjoy these as carrot muffins or frost them to make cupcakes by following the steps below.