Preheat oven to 375 degrees Fahrenheit. Butter a 9-inch loaf pan.
Toast the walnuts: Place the nuts in a dry skillet (no oil) over medium heat. Heat, shaking the pan and stirring often with a wooden spoon, until the nuts are fragrant and slightly darker brown, about 4 to 5 minutes. Stay at the stove because the nuts can easily burn! Remove immediately from the heat and transfer to a plate to stop the cooking. Then chop the walnuts: it should be enough for about 3/4 cup.
Make the batter: Mash the bananas in the bottom of a large bowl, enough for 1 1/3 cup. Whisk in the oil, brown sugar, egg, and vanilla extract.
In a separate bowl, mix the all purpose flour, whole wheat flour, baking soda, cinnamon, allspice, and kosher salt. Gradually mix the wet ingredients into the dry ingredients with a spatula, taking care not to overmix. Gently fold in the 1/2 cup walnuts.
Bake: Pour the batter into the loaf pan and top with the remaining 1/4 cup chopped walnuts. Bake 45 minutes, then add foil loosely on the top. Bake another 10 to 15 minutes until a toothpick inserted into the center of the loaf comes out clean (or the internal temperature is 200 degrees): the exact timing will depend on the size of the pan.
Cool: Cool in the pan for 15 minutes. Then run a knife around the edge and invert the loaf onto a cooling rack. Serve warm or at room temperature. Storage info: The bread lasts 4 to 5 days at room temperature wrapped in aluminum foil, refrigerated for 10 days, or frozen for 3 months (slice it into pieces and wrap it in plastic wrap then a freezer-safe bag or container).