Preheat the oven to 375° F. Lightly grease mini loaf pans or line with parchment paper. I recommend mini loaf pans or muffin tins for this recipe, so that they cook all the way through more easily. The flourless batter is dense, so with large loaf pans you can end up with gooey batter on the inside, no matter how long you bake it.
Mash the banana, eggs, and nut butter together until smooth.
Grate the carrot and apple and stir it into the mixture along with the oats, nuts, any additional fruit, and spices.
Bake for 25 - 30 minutes, or until a toothpick comes out clean.
Allow the bread to cool for 30 minutes and transfer to a wire rack.
Slice into 4 equal servings. Store in an airtight container or freeze.
I use instant oats because they are more absorbent than rolled oats and produce a better texture in baked goods. You can make your own by pulsing rolled oats in a food processor until they are like a coarse flour.