Preheat the oven to 375F and grease or line a 9 x 13 baking dish with parchment paper.
Add all of the dry ingredients apart from the walnuts to the baking dish, including the oats, oat flour, baking powder, cinnamon and ginger. Mix. Then add the wet ingredients, including the carrot, yogurt, maple syrup, plant milk and vanilla. Stir until fully combined.
Top with crushed walnuts if desired, and place in the oven for 50 to 55 minutes until cooked through.
While the baked oatmeal cooks, you can make the cashew cream cheese icing by adding all listed ingredients to a high speed blender or food processor. Combine until smooth, and ice the top of the oatmeal once cooled. Store the oatmeal in the refrigerator for up to 4 days.
If you don't have oat flour, you make your own in a food processor or coffee grinder. Simply grind some oats until you get a fine flour.
For a nut-free version, you can use sunflower or pumpkin seeds. As for the frosting, you can blend up some plain tofu instead of the cashews.
It's possible to make a similar baked oatmeal dish using zucchini. Bear in mind, zucchini has a higher water content, so you may want to squeeze the moisture out before adding zucchini into the pan.
Generally speaking, you can use any type of vegan yogurt, as long as it's unsweetened. I used coconut yogurt because it tends to be thick and creamy.