Preheat your oven to 350°F (175°C). Grease a muffin tin or line with muffin liners.
In a large bowl, mix together the grated carrots, mashed bananas, and brown sugar.
Add the vegetable oil, beaten egg, and vanilla extract. Mix well.
In another bowl, whisk together the whole wheat flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Spoon the batter into the prepared muffin tin, filling each cup about ⅔ full.
Bake for 20-25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Let the muffins cool in the tin for 5 minutes, then transfer to a wire rack to cool completely.