Preheat the oven to 350ºF. Line a baking sheet with parchment or a Silpat.
In a medium bowl, whisk together flour, oats, baking soda, and cinnamon.
In the bowl of a stand mixer, combine butter and sugar until creamy.
Add banana, squash, and egg and mix until combined.
Add dry ingredients and mix until just incorporated.
Drop tablespoonfuls of dough onto the prepared baking sheet and bake 12-15 minutes or until the cookies are just set. Allow to cool on the pan 4-5 minutes, then transfer to a cooling rack to cool completely.
Cinnamon Frosting
Once the cookies are completely cool, prepare the frosting. In a small bowl, combine all the ingredients and whisk until no sugar lumps remain.
If you prefer a smoother glaze for drizzling, add a few drops of milk until it reaches the desired consistency.