Make the dough: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, brown sugar, and vanilla extract. Mix on low until combined and there are no butter chunks.
226 grams unsalted butter, 113 grams light brown sugar, 2 teaspoons vanilla extract
Add in the sourdough discard, peanut butter, flour, salt and oats and mix to combine into a stiff cookie dough.
57 grams peanut butter, 43 grams sourdough discard, 255 grams all purpose flour, ½ teaspoon salt, 50 grams rolled oats
Chill the dough: On a large piece of plastic wrap, turn out the dough and form into a log that is about 15-inches long. Wrap tightly in plastic wrap and refrigerate for at least 4 hours or overnight.
Prep: Preheat the oven to 350°F. Line two baking sheets with parchment paper.
Remove the dough from the fridge and cut the log into 44 cookies that are about ¼-inch thick. Place on the prepared baking sheets. Poke the tops of the cookies with a fork a couple times.
Bake: Bake for 22 to 24 minutes or until golden brown around the edges. Allow the cookies to cool completely on the baking sheets. Then turn over every other cookie and pair them together.
Make the filling: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, powdered sugar, peanut butter, vanilla, salt and milk as needed. Mix until combined and then speed mixer up to high and mix for 1 to 2 minutes or until light and fluffy.
113 grams unsalted butter, 226 grams powdered sugar, 57 grams peanut butter, ½ teaspoon vanilla extract, ¼ teaspoon salt, 14 grams milk
Assemble the cookies: Transfer the filling to a piping bag and cut a hole. Pipe dollops of frosting onto the top of the turned over cookies and sandwich them together.
Store in a cool, dry place for up to 5 days.
High Altitude — Bake at 350°F for 20 to 22 minutes or until golden brown around the edges.