Preheat your oven to 350°F (175°C). Lightly grease and flour a 9x5 inch loaf pan, or line it with parchment paper leaving an overhang on the sides for easy removal.

In a large mixing bowl, whisk together the all-purpose flour, baking soda, baking powder, salt, ground cinnamon, and ground nutmeg until well combined. Set aside.

In a separate medium mixing bowl, mash the ripe bananas thoroughly with a fork until mostly smooth but with a few small lumps remaining. Add the eggs, granulated sugar, brown sugar, vegetable oil (or melted butter), and vanilla extract to the mashed bananas. Whisk until the mixture is well combined and smooth.

Pour the wet ingredients into the dry ingredients. Using a rubber spatula, gently fold until just combined. Be careful not to overmix; a few streaks of flour are acceptable.

Gently fold in the finely shredded carrots and optional chopped walnuts or pecans until evenly distributed.

Pour the batter into the prepared loaf pan and spread evenly.

Bake for 50-60 minutes, or until a wooden skewer or toothpick inserted into the center of the loaf comes out clean. If the top starts to brown too quickly, you can tent it loosely with aluminum foil after 35-40 minutes.

Remove the loaf from the oven and let it cool in the pan for 10-15 minutes before carefully transferring it to a wire rack to cool completely. Slicing while warm is fine, but it holds its shape better when fully cooled.