Preheat the oven to 350°F (175°C). Grease a 9×5 inch loaf pan or line it with parchment paper.
In a large mixing bowl, mash the ripe bananas with a fork until smooth.
Add the eggs, unsweetened applesauce, granulated sugar, brown sugar, and vanilla extract to the mashed bananas. Mix well until combined.
In a separate bowl, whisk together the gluten-free all-purpose flour blend, 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.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake in the preheated oven for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and let the banana bread cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing.