Adjust an oven rack to the lower-middle position and heat the oven to 350 degrees F (175 degrees C) degrees. Grease the bottom and sides of a 9 by 5-inch loaf pan; dust with flour, tapping out the excess.
Spread the walnuts on a baking sheet and toast until fragrant, 5 to 10 minutes. Set aside to cool.
Whisk the flour, sugar, baking soda, salt, and walnuts together in a large bowl; set aside.
Mix the mashed bananas, yogurt, eggs, butter, and vanilla with a wooden spoon in a medium bowl. Lightly fold the banana mixture into the dry ingredients with a rubber spatula until just combined and the batter looks thick and chunky. Scrape the batter into the prepared loaf pan.
Bake until the loaf is golden brown and a toothpick inserted in the center comes out clean, about 55 minutes. Cool in the pan for 5 minutes, then transfer to a wire rack. Serve warm or at room temperature. (The bread can be wrapped with plastic wrap and stored at room temperature for up to 3 days.).