To 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.
Add in the sourdough discard, peanut butter, flour, and oats and mix to combine into a stiff cookie 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.
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 of times.
Bake for 20 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.
To make the filling: In the bowl of a stand mixer fitted with the paddle attachment, add the butter, powdered sugar, peanut butter, vanilla, 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.
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.