Preheat the oven to 350°F (175°C) and line a 12-cup muffin tin with paper liners.
In a large mixing bowl, combine the mashed bananas, eggs, unsweetened applesauce, granulated sugar, and vanilla extract. Mix well until smooth.
In a separate bowl, whisk together the gluten-free all-purpose flour, baking soda, baking powder, salt, and ground cinnamon.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Divide the batter evenly among the 12 muffin cups, filling each about 3/4 full.
Bake in the preheated oven for 18 to 20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the tin for 5 minutes.
Transfer the muffins to a wire rack to cool completely before serving.