Preheat the oven to 375° F. Line two standard muffin pans with paper liners. In the bowl of a stand mixer fitted with the wire whisk, combine the bananas, yogurt, sweet potatoes, brown sugar and white sugar. Mix on high speed until lightened in color and texture, about 3 minutes. Add the butter, eggs and vanilla and mix until well-combined, stopping to scrape down the sides of the bowl when necessary.
In a medium bowl, combine the all-purpose flour, white whole wheat flour, salt, baking soda, cinnamon and nutmeg. Add the dry ingredients to the wet ingredients and mix on low speed until just combined.
Spoon the batter into the muffin cups, filling them a little more than halfway. Bake for 18-20 minutes, until a tester inserted into the center of a muffin comes out clean.