This cake is friendly to substitutes (see Substitutions, below). Bananas stand in for eggs, both binding the batter and giving it body — use bananas whose peels have gone mostly black for the best flavor. Since you’ll want to finish this cake with lime zest, grate it into a bowl before you juice it for the cake batter, since zest is easier to remove from whole, intact limes. Though the amount of sugar here may seem tedious, it gives the cake the best balance of sweetness for my taste. But if you want a less sweet cake, use ¾ cup total; for a sweeter crowd-pleaser, use 1 cup. Working within this small range will not affect the cake texture or chemistry negatively.
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Start making the frosting: In a small nonstick skillet or saucepan, whisk together the sugar, flour and salt, then pour in the milk and stir until smooth. Place the pan over medium-high heat and bring to a boil, stirring steadily with a small rubber spatula, as the mixture thickens in spots. Continue cooking until the mixture is fully smooth and thickened to the consistency of peanut butter (the paste will pull away from the side of the skillet too). Remove the pan from the heat and scrape the pudding into a medium bowl, preferably one with high sides. Let the pudding cool to room temperature while you make the cake.
Make the cake: Heat the oven to 350 degrees. Grease an 8-inch (or 9-inch) round cake pan with oil, then line the bottom with a round of parchment paper. Grease the paper, then sprinkle the inside of the pan with all-purpose flour, turning to coat evenly and tapping out any excess.
In a medium bowl, whisk together both flours, the sugar, cinnamon, salt, baking powder and baking soda. In another large bowl, whisk together the oil, bananas, oat milk, lime juice and vanilla. Pour the liquid ingredients over the dry ingredients, then stir just until the batter comes together. Add the carrots and, if using, walnuts and/or raisins, and stir until everything is evenly coated in batter.
Scrape the batter into the prepared pan and smooth the top. Bake until golden brown and a toothpick inserted in the center of the cakes comes out clean, about 45 minutes for an 8-inch cake (35 to 37 minutes for a 9-inch cake). Transfer the pan to a wire rack and let cool for 10 minutes. Invert the pan onto the rack, letting the cake fall out, then remove and discard the parchment paper round. Turn the cake right side up and let cool completely.
Once the cake is cool, finish making the frosting: Add a spoonful of the shortening to the cooled pudding and stir with a wooden spoon until smooth. Add another spoonful and stir once more. Add the remaining shortening and the vanilla, stir until smooth, then vigorously beat with the wooden spoon until the frosting is light and fluffy, 10 to 20 seconds.
Scrape the frosting onto the cooled cake and spread it evenly over the top. Sprinkle the grated lime zest over the top of the cake before serving.
No whole wheat flour? Use spelt, buckwheat, oat or rye flour.
No oat or almond milk? Use soy, rice or regular whole dairy milk if you’re unconcerned with keeping the recipe vegan.
No lime juice? Use lemon juice (do not use orange juice).
No lime zest? Use lemon or orange zest.
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