Preheat oven to 350 degrees F.
Lightly grease a 9 x 13-inch baking pan.
Mix soymilk in a measuring cup with vinegar and set aside for 5 minutes or more to thicken.
In a large bowl, mix together oil, mashed bananas, sugar, and vanilla.
In another bowl, whisk or sift together, flour, baking powder, baking soda, salt, and nutmeg.
Next, mix 1/3 of the dry ingredients into the wet ingredients.
Then mix in 1/2 of the milk mixture.
Then add another 1/3 of the dry ingredients and then the rest of the milk mixture, and lastly add the rest of the dry ingredients, along with the nuts, blending well.
Pour batter into a 9 x 13-inch lightly greased cake pan and bake in the preheated oven for approximately 30-35 minutes, checking with a toothpick for no sign of wet batter.
Let cool on a wire rack before frosting.
