Preheat your oven to 350°F (175°C) and grease a round cake pan or a rectangular baking dish. You can also use cupcake molds for individual servings.
In a large mixing bowl, whisk together the eggs, unsweetened applesauce, coconut oil, and peanut butter until well combined.
Gradually add the oat flour and baking soda to the wet ingredients, stirring until everything is well combined and forms a thick batter.
Fold in the grated carrots into the batter until evenly distributed.
Pour the batter into the greased pan or divide it among the cupcake molds.
Bake in the preheated oven for about 25 minutes for a full cake or 20 minutes for cupcakes. Keep a close eye on it, as baking time may vary depending on your oven and the thickness of the cake.
To check if the cake is done, insert a toothpick into the center. If it comes out clean, the cake is ready.
Allow the cake to cool completely before frosting and serving to your dog.