Preheat an oven to 325 degrees F (165 degrees C). Grease two 4x8-inch loaf pans.
Mix the cider vinegar and milk together in a small bowl; set aside. Cream together the butter and sugar in a large bowl with an electric mixer. Beat in the sliced bananas until well incorporated, then stir in the milk mixture. Mix in the flour and baking soda until just combined, then mix in the walnuts and eggs until evenly distributed. Divide the batter evenly between the two prepared pans. Place the pans on a baking sheet for easier handling.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack. Wrap in plastic wrap and store in refrigerator.
