Preheat oven to 350 degrees F (175 degrees C). Grease a 9x5-inch loaf pan. In a bowl, whisk together the whole wheat flour, all-purpose flour, baking soda, kosher salt, cinnamon, and nutmeg.

Mash together the butter and sugar in a mixing bowl until creamy; beat in the egg and vanilla extract with an electric beater. In a third bowl, mash the bananas with the yogurt until smooth. Mix the flour mixture into the butter mixture in several additions, alternating with the banana mixture until the batter is smooth. Pour into the prepared loaf pan.

Bake in the preheated oven until a toothpick inserted into the center of the banana bread comes out clean, 50 to 60 minutes. Let the bread cool in the pan for 10 minutes, then finish cooling on a cooling rack.