Preheat oven to 350°F.
In a medium bowl, smash the banana with a potato smasher. Add peanut butter and egg and whisk vigorously to combine.
Add 1 cup of flour to the mixture and mix with a wooden spoon to form dough.
Add 1/2 cup of rolled oats and combine with the dough, knead the rolled oats into the dough using the spoon or your hand.
At this point, start to add more flour, up to 1/2 cup to achieve desired dough consistency, something between pie and bread dough, not too sticky.
Sprinkle some flour on a cutting board and turn the dough out, knead and shape into a disk.
Roll dough out to about 1/4" thick. Cut with a 2” bone shape cookie cutter, and place on a parchment paper lined baking sheet about 1/2” apart.
Reshape scraps and roll out again to 1/4” thick and continue cutting out treats until finished. This will yield approximately 2 dozens.
Bake for 40 minutes, flip treats half way through baking time.
Turn off the oven when done and leave them in the oven to dry overnight.
Store in an airtight container or Ziploc bag. If the treats are baked until they really dry out, they could last outside for a while but I always store them in the fridge just to be safe.