Preheat oven to 350°F (175°C). Line a small cupcake pan with paper liners or lightly grease with coconut oil.
Mash the banana thoroughly in a mixing bowl until smooth with only small lumps.
Add the egg and mix well. Stir in peanut butter and yogurt until uniform.
Add rolled oats and stir until coated. Mix in water or broth to loosen batter to a thick muffin batter consistency.
Spoon batter into cupcake liners, filling each about three-quarters full.
Bake for 18–22 minutes until tops are firm and slightly golden. A toothpick inserted in center should come out mostly clean.
Cool in pan for 10 minutes, then transfer to a rack to cool completely.
Optional: top cooled cupcakes with a thin layer of plain yogurt or mashed banana.
Serve one cupcake as a treat (half for small dogs). Store leftovers in an airtight container in the refrigerator for up to 5 days.