Preheat the oven to 350 degrees F (175 degrees C). Grease and flour two 8-inch round pans.
Whisk together flour, baking soda, and salt in a small bowl; set aside.
Cream white sugar, brown sugar, and butter in a large bowl until light and fluffy. Beat in eggs, one at a time. Mix in mashed bananas. Add flour mixture alternately with buttermilk, mixing well after each addition. Stir in chopped walnuts. Pour batter into the prepared pans.
Bake in the preheated oven until cake springs back when gently pressed, about 30 minutes.
Remove cake layers from the oven and place them on a damp tea towel to cool.
