Preheat your oven to 350°F (175°C). Grease and flour a 6-inch round cake pan, or line it with parchment paper.
In a large mixing bowl, whisk together the flour and baking soda. If using grated carrots or mashed banana, add them now.
In a separate medium bowl, combine the unsweetened applesauce, peanut butter, oil, water, and egg. Whisk until well combined.
Pour the wet ingredients into the dry ingredients. Mix until just combined. Be careful not to overmix; a few lumps are okay.
Pour the batter evenly into the prepared cake pan.
Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for about 10 minutes before inverting it onto a wire rack to cool completely.
While the cake cools, prepare the frosting. In a small bowl, mix the Greek yogurt and peanut butter until smooth. Chill until ready to frost.
Once the cake is completely cool, spread the frosting evenly over the top and sides. Decorate with dog-safe toppings if desired (e.g., a few blueberries, a sprinkle of shredded carrot).