Preheat your oven to 350°F (175°C) and line a muffin tin with muffin liners or grease it lightly.
In a large mixing bowl, combine the pumpkin puree, mashed bananas, granulated sugar, brown sugar, vegetable oil, eggs, and vanilla extract. Whisk until the mixture is smooth and well combined.
In another bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt.
Gradually fold the dry ingredients into the wet ingredients using a rubber spatula. Mix until just combined; be careful not to overmix. If desired, gently fold in the chopped walnuts or chocolate chips.
Scoop the batter evenly into the prepared muffin tin, filling each cup about two-thirds full.
Bake in the preheated oven for 20-25 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Once baked, remove the muffins from the oven and allow them to cool in the tin for 5 minutes. Then transfer them to a cooling rack to cool completely.