Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper.
Mash the banana in a mixing bowl until smooth and creamy.
Grate the carrot finely using a box grater.
Add the egg, coconut oil, and optional peanut butter to the mashed banana. Stir until smooth and glossy.
Stir in the rolled oats, oat flour, and grated carrot. Mix until a soft dough forms. If sticky, add extra oat flour; if too stiff, add a teaspoon of water.
Dust a countertop with oat flour and roll or press the dough to 1/4 inch thickness.
Cut into small squares, circles, or bone shapes. Place on the prepared baking sheet with small gaps between treats.
Bake for 18–22 minutes until edges are lightly golden and treats feel firm. For crunchier treats, leave in the oven with heat off for 5 more minutes.
Cool completely before serving. Store in an airtight container in the refrigerator for up to 7 days or freeze for up to 3 months.