Mix chocolate cake mix, 1 1/4 cups water, 1/2 cup olive oil, and 3 egg replacers in a large bowl with an electric mixer on medium speed until well combined. Pour into one of the prepared baking dishes.
Mix white/yellow cake mix, 1 cup water, 1/2 cup vegetable oil, and 3 egg whites in another large bowl on medium speed with clean beaters until well combined. Pour into the second prepared baking dish.
Bake cakes in the preheated oven until a toothpick inserted in the center comes out clean, 30-35 minutes. Remove from the oven and let cool to room temperature on a wire rack, about 30 minutes.
Once the pudding has thickened, remove the saucepan from the heat and stir in the vanilla extract and a pinch of salt.
Pour the hot pudding into individual serving dishes or a large bowl.
Cover the pudding with plastic wrap, making sure the plastic wrap touches the surface of the pudding to prevent a skin from forming.
Refrigerate the pudding for at least 2-3 hours or until it's fully chilled and set.