Preheat the oven to 425°F. Line a muffin tin with paper liners or lightly grease the pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon.
In a large bowl, whisk the brown sugar, melted butter, and vegetable oil until smooth.
Add the eggs and vanilla extract, whisking until fully combined.
Stir in the Greek yogurt and mashed bananas until the mixture is smooth.
Fold in the grated carrots.
Add the dry ingredients to the wet ingredients and mix just until combined. Do not overmix.
Fold in the chopped nuts, if using.
Divide the batter evenly between the muffin cups, filling them almost to the top for tall bakery-style muffins.
Sprinkle the tops with coarse sugar or additional nuts if desired.
Bake at 425°F for 5 minutes, then reduce the oven temperature to 350°F without opening the oven. Continue baking for 12–15 minutes, or until a toothpick inserted into the center comes out clean.
Allow muffins to cool in the pan for 5 minutes, then transfer to a wire rack to cool completely.