In a small bowl, mix together the banana, peanut butter and egg; mix until completely combined. To a medium bowl, add the flour and rolled oats; mix. Add the wet ingredients to the dry ingredients and combine until the dough comes together.
Transfer the dough to a piece of heavily floured wax paper. Note: The dough will be sticky so flouring everything (wax paper, rolling pin and cookie cutter) is important. Roll the dough to a 1/4-inch thickness and cut out the cookies using a cookie cutter of choice. Re-roll the scraps and cut out more cookies. (I used a small heart cookie cutter and ended up with 22 cookies.) Transfer the cookies to a parchment-lined baking sheet. These cookies won't spread so placing them close together is okay. If you don't want to roll the dough out, alternatively, you can scoop teaspoons of dough onto a baking sheet, flattening the cookies with the ball of your palm.
Bake the cookies for 30-40 minutes or until golden brown. Allow the cookies to come to room temperature on a cooling rack. Cookies will be good for up to 2 weeks in an airtight container.
*Oats are technically a grain. I added them for texture. You can of course eliminate this and you'll be good to go.