Preheat oven to 350°F. Spray an 8×4-inch loaf pan with nonstick spray.
In a stainless steel skillet (or another light-bottomed pan) over medium-high heat, melt the butter, swirling the pan occasionally. The butter will begin to foam, which signals the water in the butter is boiling off. (Watch closely at this stage; browned butter and burnt butter are within seconds of each other.) Tiny brown specs of milk solids will begin to appear at the bottom of the pan; constantly stir and scrape these off using a spatula so they don't stick. As soon as the butter turns chestnut brown and smells nutty, remove the pan from the heat.
In a medium bowl, sift the flour, baking soda, baking powder, and salt. Set aside.
In a large bowl, mash bananas with a fork. (You can use up to 5 regular-sized bananas or 6 small bananas in this recipe.)
To the bananas, add browned butter, sugar, egg, and vanilla. Mix well.
Add the dry ingredients, mixing until ingredients are JUST incorporated. (Be careful not to overmix.)
Gently fold in ¾ cup of chopped walnuts.
Pour the batter into the prepared loaf pan. Top with the remaining ¼ cup walnuts.
Bake for 35 minutes.
After 35 minutes, cover with aluminum foil. Bake for an additional 30-35 minutes, or until a toothpick comes out with crumbs, but not wet batter.
Let the bread cool in the pan for about 15 minutes before transferring it to a wire rack or plate to cool completely. Slice and enjoy!