Preheat oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone baking mats. (Always recommended for cookies.) Set aside.

Whisk the flour, oats, baking soda, baking powder, and salt together until combined. Set aside.

In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the butter, brown sugar, and granulated sugar together on high speed until completely smooth and creamy, about 2 minutes. Add the peanut butter, egg, and vanilla extract and beat on high speed until combined, about 1 minute. Scrape down the sides and bottom of the bowl as needed to combine. Add the dry ingredients to the wet ingredients and mix on low until combined.

Scoop cookie dough into balls, about 2 heaping teaspoons of dough per cookie (almost 1 Tablespoon—any size around here is great), and arrange 3 inches apart on the baking sheets. Using the back of a measuring cup or drinking glass, gently press down on each dough ball to slightly flatten. Bake for 10-11 minutes or until edges are lightly browned and appear set.

Allow cookies to cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely before sandwiching.

Make the frosting: In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the frosting ingredients together until combined and smooth. Spread the frosting onto the flat side of one cookie and sandwich with the other. (Or you can use a piping tip—I used piping tip #12 to pipe the frosting onto the cookies before sandwiching.) Repeat with remaining.

Cookies will stay fresh covered at room temperature for 3 days or in the refrigerator for 1 week.