Make the pumpkin-banana purée. Preheat the oven to 350°F with a rack positioned in the center. Line a baking sheet with parchment paper. In a small bowl, mash and combine the pumpkin purée and banana until mostly smooth. Spread the mixture flat onto the parchment paper-lined baking sheet. Bake on the center rack for 45 minutes.
Remove the pumpkin-banana mixture from the oven and allow to cool completely. To speed up the cooling process, I spread the mixture around the sides of a mixing bowl and set it in the freezer for about 10 minutes. I also set the baking tray aside for my cookie dough balls later.
Mix the dry ingredients and cinnamon-sugar. In a small bowl, whisk together the dry ingredients for about 30 seconds until well-mixed: all-purpose flour, whole-wheat flour, table salt, baking soda, and baking powder. Set the bowl aside. In another small bowl with a wide mouth, mix together the cinnamon-sugar and set it aside for later.
Prepare the dough. To a stand mixer fitted with the paddle attachment, add the butter. Mix on medium-high speed (KitchenAid 6) for 20 seconds. Scrape the paddle and bowl with a silicone spatula. Mix for another 20 seconds on medium-high. Scrape and mix again. The butter should be nicely soft and ready for beating.
Add the brown and white sugars to the mixer bowl. Beat at medium-low speed (KitchenAid 4) for 2 minutes. Stop the mixer. Scrape the bowl and paddle. Continue beating at medium-high speed (KitchenAid 6) for an additional 3 minutes. The mixture should be lighter in color and fluffed.
Add the roasted and cooled pumpkin-banana mixture to the stand mixer bowl. Beat at medium speed (KitchenAid 5) for 2 minutes.
Stop the mixer. Scrape the bowl and paddle. Add both eggs. Continue beating for 3 minutes at medium speed. Stop the mixer. Scrape the bowl and paddle. Add the vanilla. Continue beating for 2 minutes at medium speed.
Stop the mixer. Scrape the bowl and paddle. Add the dry ingredient mixture. Mix on the lowest speed for 25-35 seconds, keeping an eye out for almost all the flour to be mixed in. Remove the bowl from the mixer and incorporate the rest of the flour in with a silicone spatula.
Roll the cookies in cinnamon sugar. Bring your bowl of cinnamon sugar nearby. Using a kitchen scale or a lightly oiled ¼-cup measuring cup, scoop 86-gram dough balls into your hand and shape the dough into a ball. Immediately roll the ball in cinnamon-sugar, place it on a baking tray (the same pan used for the pumpkin-banana purée, if you’d like!), and pat the dough ball into a tall puck shape. Repeat until all the dough is scooped into balls, rolled, and shaped on the tray.
Cover the cookies loosely and chill in the refrigerator. You can make the cookies now or continue to chill in the fridge for up to 72 hours. You can also tightly wrap the dough balls and freeze for up to 3 months.
When you’re ready to bake the cookies, preheat the oven to 375°F for at least 20 minutes. I like to use an oven thermometer to make sure the internal temperature is correct.
Place the cookies at least 2 inches apart on a parchment-lined cookie sheet. I use a silicone baking mat. Bake for 15 minutes. If you prefer a slightly more underbaked cookie: 14 minutes. A fully cooked and cakey cookie: 16 minutes.
The cookies will bake into domes, then sink in the center while cooling. Enjoy warm with a glass of milk or a scoop of ice cream with caramel drizzle. The cookies also keep well for days at room temperature.