Wash, dry, and peel the carrots. Then, use a box grater or food processor to grate them.
Add the bananas to a large bowl and thoroughly mash them with a fork or potato masher. Then, add the eggs, unsweetened applesauce, vanilla extract, and grated carrots to the bowl with the mashed bananas. Mix these ingredients.
In a separate large bowl, combine the flour, sugar, cinnamon, nutmeg, baking powder, baking soda, and salt.
Add the dry ingredients to the bowl with the wet ingredients a little at a time and mix. Make sure all of the ingredients are combined, but don't over-mix.
Lightly spray a nonstick silicone baking pan with oil. Then, pour the batter into the muffin pan, forming 12-14 muffins. Or, you can place muffin liners in the muffin pan and pour the batter into the liners.
Bake the muffins at 350℉ for 24-26 minutes. Once done, let the muffins cool, carefully remove them from the muffin pan, and enjoy!