Preheat the oven to 350 degrees F. Line the bottom of a 9- by 5-inch loaf pan with parchment, and butter the bottom and sides.
Whisk together the rice and tapioca flours, 2/3 cup of the sugar, baking powder, baking soda, cinnamon and salt in a medium bowl.
Whisk together the butter, egg yolks and vanilla in a large bowl. Fold in the mashed bananas.
Whip the egg whites with an electric mixer on medium-high speed in another large bowl until foamy. Add the remaining 1 tablespoon sugar, and continue to beat on medium-high until soft peaks form.
Fold the flour mixture into the banana mixture until just combined (it's OK if there are some lumps). Stir in 1/3 of the egg whites until combined, then gently fold in the rest. Transfer the batter to the prepared loaf pan, and bake until the bread is golden brown and a toothpick inserted in the center comes out clean, 55 minutes to 1 hour. Run a knife around the edges, and let cool completely in the pan on a rack. The bread can be stored at room temperature in an airtight container for up to 3 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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