Preheat your oven to 350°F and line a 12-cup muffin pan with muffin liners.
Mash the bananas in a large bowl. Add the melted butter, granulated sugar, brown sugar, sour cream, vanilla extract, and egg. Whisk until well combined.
In a separate bowl, whisk together the all-purpose flour, baking soda, cinnamon, nutmeg, and salt.
Gradually fold the dry ingredients into the wet ingredients until just combined. Be careful not to overmix.
Gently fold in the finely grated carrots until evenly distributed throughout the batter.
Divide the batter evenly among the muffin liners.
Bake for 20–25 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the muffins cool in the pan for about 5 minutes, then transfer them to a wire rack to cool completely before frosting.
While the muffins cool, beat the softened butter until smooth and creamy. Add the softened cream cheese and mix until smooth. Gradually add the powdered sugar, starting with 1 cup and adding more if desired. Mix in the vanilla extract.
Once the muffins are completely cooled, pipe or spread the cream cheese frosting over the tops.
Enjoy immediately or refrigerate until ready to serve.