Preheat your oven to 350°F and grease a 9x5 inch loaf pan or similar size with dairy-free butter or oil. While the oven heats, grate the carrots using the small holes on a box grater or a food processor, then squeeze out excess moisture by wrapping them in a clean kitchen towel and pressing firmly—this prevents the bread from becoming too wet and soggy. Mash the bananas in a separate bowl until mostly smooth with just a few small lumps remaining. Have all other ingredients measured and ready to go.
In a large mixing bowl, whisk together the egg, applesauce, mashed bananas from Step 1, maple syrup, and vanilla extract until well combined and smooth. This creates an emulsified base that will help bind all the dry ingredients and create a moist crumb.
In a separate bowl, whisk together the quick oats, whole wheat flour, baking soda, cinnamon, ginger, nutmeg, and salt. This pre-mixing ensures the leavening agent and spices are evenly distributed throughout the batter, preventing any pockets of baking soda or uneven spice distribution. I find that whisking the dry ingredients separately gives a more consistent rise and better flavor balance than mixing everything together at once.
Pour the dry ingredient mixture from Step 3 into the wet ingredients from Step 2 and fold gently with a spatula until just combined—do not overmix, as this can develop gluten and create a tough, dense texture. Once mostly combined, fold in the grated carrots from Step 1 and chopped walnuts until evenly distributed throughout the batter. The batter should be thick and chunky.
Pour the batter from Step 4 into the prepared pan from Step 1 and smooth the top with a spatula. Bake at 350°F for 30 minutes, then cover the top loosely with aluminum foil to prevent over-browning. Continue baking for 10-20 minutes more (total of 40-50 minutes) until a toothpick inserted into the center comes out with just a few moist crumbs—not completely dry. The bread will continue to cook slightly as it cools.
While the bread bakes, make the frosting by beating the softened dairy-free cream cheese and dairy-free greek yogurt together in a clean bowl until smooth and creamy. Add the powdered sugar, maple syrup, cinnamon, and vanilla extract, then beat until well combined and fluffy. Add the plant-based milk 1 tablespoon at a time until you reach a spreadable consistency—not too thick, not too thin. Once the bread comes out of the oven, let it cool in the pan for 10 minutes, then turn it out onto a wire rack to cool completely before frosting.
Once the bread from Step 5 is completely cool, generously spread the frosting from Step 6 over the top and sides. I like to let the frosting set for 15-20 minutes before slicing so it holds its shape better, but this is optional depending on how quickly you want to serve it.