Heat the oven to 350ºF. Grease a 12-cup muffin tin and set aside.
Add the egg, banana, applesauce, milk and syrup into a blender. Blend on high until smooth. Add the oats and blend again until smooth. Add remaining ingredients (minus the carrots and walnuts) and blend again until combined. Stir in carrots.
Fill each cup ¾ of the way full. Sprinkle with walnuts if using and press down slightly so they stay in place. Add water to the ones that are unused.
Bake on the center rack for 22 - 24 minutes until a cake tester inserted into the center comes out clean.
Cool in the pan for 5 minutes, then transfer to a wire rack and cool completely. If desired, reheat in the toaster oven or in the microwave (and definitely drizzle with some peanut butter!)