You can make the peanut butter filling while the dough is chilling, or while the cookies are baking and cooling.
Brown the butter - In a small saucepan, add the butter. While stirring, heat over medium heat until the butter has melted, started to turn brown with little brown flecks, and smells nutty. Remove from the heat. Note - This small amount of butter will brown quickly, so do not turn your back on it.
Add the peanut butter to the brown butter and return to low heat to just melt the butter and peanut butter together.
Transfer the peanut butter mix to a medium mixing bowl. Using a hand held mixer with beater attachments, add the powdered sugar in a few increments. After each addition, mix on low speed to just incorporate the powdered sugar (and not make a mess!). After the final addition, turn up to medium speed to mix the filling until smooth and creamy. Tip - If you prefer a thicker filling, add more powdered sugar a tablespoon at a time. Keep in mind, the filling will thicken as it cools.
Cover the bowl with a piece of plastic wrap and set aside on the counter until ready to assemble the cookies.
Bake the Cookies - 15 minutes into the dough chill time, preheat the oven to 350F degrees and adjust the oven rack to the middle position. Line two baking sheets with pieces of parchment paper.
Place the chilled cookie dough between two sheets of parchment paper. Roll the dough to about 1/4 inch thick - or thinner if you prefer a thinner cookie. Tip - Use the sheets of parchment from the lined baking sheets to save on parchment! You will need to re-roll and cut out the dough at least twice to use up the dough.
Use a round cookie cutter (I used a 2-inch round cookie cutter) and cut out the cookies. Place the cut out cookies on the baking sheets (put the parchment back on the baking sheet!), about 2 inches apart. Note - These cookies do not spread. Spacing them apart allows for better heat destitution around the cookies. You will likely have scraps left that are too small to cut into a cookie that can be discarded or baked.
The number of cookies you get from the dough will depend on how thick/thin you roll the dough, and the size of cookie cutter.
Bake the cookies, one baking sheet at a time, for 8 minutes. Cool the cookies on the baking sheet set on a wire cooling rack for 5 minutes. Then transfer the cookies to the rack to cool completely. Note - Because of the dark color, it will be hard to tell if the cookies are baked to golden perfection. You can test if the cookies are done by gently touching them and they have a baked exterior, and you will start to smell the baked cocoa. Do not overbake the cookies as they can burn very quickly.
Assemble the Cookies - On a baking sheet, divide the cookies equally into tops and bottoms. Pipe or spoon equal amounts of the peanut butter filling onto the cookie bottoms. Put the cookie tops on. Chill the cookie sandwiches for at least 30 minutes in the refrigerator or freezer. Note - Chilling the assembled cookies makes them easier to dip into the melted chocolate.