Mix the nondairy milk with the apple cider vinegar and set aside for a minute.
Combine the flour, baking soda, baking powder and salt in a bowl.
Beat together the oil and sugar with a handheld mixer or in a stand mixer or with a whisk for about two minutes.
Add the nondairy milk and beat until just mixed.
Add the mashed bananas and applesauce and beat in. Make sure there are no big lumps.
Add the flour to the sugar-banana mixture in two batches, mixing well after each addition until the mixture is smooth. Don't overdo it.
Line the bottom of two 9-inch cake pans with parchment or wax paper. Oil and flour the pans.
Divide the batter equally between the two pans and bake 25-30 minutes in a 350-degree oven until a toothpick inserted in the center of each cake comes out clean.
Place on a rack to cool, about 10 minutes. Unmold, peel off the parchment paper, and leave on the rack to cool thoroughly before frosting.