Preheat oven to 350°F (177°C). Line two large baking sheets with parchment paper or silicone mats.
In a medium bowl, whisk together flour, oats, baking soda, baking powder, and salt. Set aside.
In a large bowl (or stand mixer with paddle attachment), beat the butter, brown sugar, and granulated sugar on high speed for 2 minutes until smooth and creamy.
Mix in the peanut butter, egg, and vanilla extract until fully combined, about 1 minute. Scrape down sides of the bowl as needed.
Add the dry ingredients to the wet and mix on low until a soft dough forms.
Scoop dough into small balls (about 2 heaping teaspoons each) and arrange 3 inches apart on baking sheets. Gently flatten each one using the bottom of a glass.
Bake for 10–11 minutes, until the edges are lightly golden and set. Cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
In a clean bowl, beat together the frosting ingredients until creamy and smooth.
Spread a layer of peanut butter frosting on the flat side of one cookie and sandwich with another. Repeat for all cookies.
Keep covered at room temperature for up to 3 days or refrigerated for up to 1 week.