Preheat oven to 350 degrees. Line baking sheets with parchment paper.
Combine brown sugar, white sugar, and butter in a large bowl. Beat until creamy. Add peanut butter, milk, egg, and vanilla. Mix until well blended.
Add baking soda, salt, and flour to the creamed mixture at low speed.
Use a 2 Tbsp. sized cookie scoop to scoop cookie dough onto the baking sheet. Flatten slightly in a crisscross pattern with a fork (if the dough sticks to the fork you can dip the fork in water first).
Bake one baking sheet at a time for 10-11 minutes or until set and just beginning to brown. While cookies are still hot, shape the edges using a round cookie cutter or spatula so that they are uniform in size.
Cool completely on baking sheets.
Beat butter, powdered sugar, milk, and vanilla together until fluffy. Stir in marshmallow creme.
Pipe or spread frosting onto half of the cookies.Place the remaining cookies on top of the frosted cookies to create cookie sandwiches.
Video
Notes
I bake with light-colored metal baking sheets. If you bake on dark metal you may need to adjust the baking time so that the bottoms of the cookies don't get too dark.