Preheat oven to 350 degrees F. Line a large baking sheet with parchment paper or a Silpat and set aside.
In a small bowl, whisk together the flour, baking soda, baking powder, and salt. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, peanut butter and both sugars on medium speed until light and fluffy, about 3 minutes. Add the egg and vanilla extract. Mix until combined. With the mixer on low speed, slowly add the dry ingredients, mixing just until combined. Finally, add the oats and mix until just combined.
Take about 1 tablespoon of dough and roll into a cookie ball. Place on prepared baking sheet, leaving about 2 inches between the cookies. Bake for 10-12 minutes, or until the cookies are slightly golden and the edges are set. Let the cookies cool on the baking sheet for 5 minutes then transfer them to a wire rack to cool completely.
While the cookies are cooling, make the filling. Combine the butter, peanut butter and powdered sugar in the bowl of a stand mixer fitted with the paddle attachment. Beat until smooth. Add the heavy cream and beat until light and fluffy. Using a knife, spread peanut butter filling onto the flat side of a cookie. Find a cookie the same size and sandwich the cookies together, pressing the filling to the edges.