Preheat the oven to 350 degrees F. Line a large baking sheet with a Silpat baking mat or parchment paper and set aside.
In a medium bowl, whisk together flour, baking soda, cinnamon, allspice, and salt. Set aside.
In the bowl of a stand mixer, beat the butter and sugars together until smooth and creamy. Add in the peanut butter and mix again until smooth. Add the egg and vanilla extract and mix until combined.
With the mixer on low, slowly add in the flour mixture. Mix until just combined. Stir in the oats and diced apples.
Roll cookie dough into tablespoon size balls and place on prepared cookie sheet. Bake for 11-12 minutes or until light golden brown around the edges. Let cookies cool on the baking sheet for two minutes. Transfer to a cooling rack and cool completely.
Note-bake these cookies immediately. Do not chill the dough in the fridge. If you do, the batter will become too moist from the apples and the cookies will run!