Place the applesauce in a medium saucepan over medium heat. Bring to a low boil and then simmer, stirring occasionally, until the amount is reduced by half (to 1/3 cup/80g). On my stove, this takes about 10 minutes. Transfer the applesauce to a heat-proof bowl and set aside to cool to room temperature.

Preheat the oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats.

Whisk the oats, flour, baking soda, salt, cinnamon, allspice, and nutmeg together in a large bowl. Set aside.

Using a handheld or stand mixer fitted with a paddle attachment, beat the butter, granulated sugar, and brown sugar on medium speed until creamed together, about 5 minutes. Scrape down the sides and bottom of the bowl with a silicone spatula as needed. Add the reduced applesauce, egg yolk, and vanilla, and beat on medium-high speed until combined. Scrape down the sides and bottom of the bowl as needed.

Add in the dry ingredients and beat on low speed until combined. Finally, beat in the chopped apples and walnuts (if using). The cookie dough will be thick and sticky.

Using a medium cookie scoop, scoop mounds of cookie dough (about 1.5 Tbsp/35g of dough each) and arrange 3 inches apart on the prepared baking sheets.