For the cookies:
1/2 cup butter, at room temperature
1/2 cup shortening
1 cup brown sugar
3/4 cup granulated sugar
1 cup peanut butter
1 tsp vanilla extract
1 1/2 cups flour
2 tsps baking soda
1 tsp kosher salt
1 1/2 cups oats (quick or old fashioned)
For the frosting:
4 tbsp butter, at room temperature
1/2 cup creamy peanut butter
1 cup powdered sugar
2 tbsp milk or cream
1 tsp vanilla
Beat together butter & shortening for 2 minutes, scraping sides & bottom.
Add brown sugar & granulated sugar. Beat for 2 minutes, until light & fluffy, scraping the sides & bottom. Add 1 cup peanut butter & beat until incorporated. Add eggs & vanilla extract, beat well.
Add flour, baking soda, & salt, stir together until mostly combined. Add oats & stir until combined, scraping bottom & sides, & careful not to over mix. Cover the bowl & chill for a minimum of 3 hours or overnight.
Preheat oven to 350 degrees. Scoop out 1 or 2 inch, depending on preferred size. Bake for 8-10 minutes, ensuring middle is cooked, but careful not to over bake. Allow to cool on sheet for 5 minutes before transferring to a cooling rack. Cool completely before frosting.
Prepare the frosting. Beat butter & peanut butter together until smooth & creamy. Add powdered sugar & beat until well incorporated. Add milk, cream, & vanilla extract. Add more milk/ cream, if desired.
Spread frosting on bottom of cookie & sandwich with another.