Make the cookies: Heat the oven to 350 degrees Fahrenheit. Line two baking sheets with parchment.
In a medium bowl, sift together the two flours, baking soda, and salt. Set aside.
In the bowl of an electric mixer, cream the butter, peanut butter, and sugars with the paddle attachment until light and fluffy.
Add the vanilla and egg; continue creaming until smooth and fluffy, about 3 minutes with an electric mixer (longer by hand).
Stir in the flour mixture by hand just until it's incorporated; don't over-mix or the cookies will be tough.
Drop heaping tablespoonfuls of batter, spaced about 2 inches apart, onto the lined baking sheets. With floured fingers, flatten each dab of batter into a 2-inch round.
Bake until the cookies are puffed and golden, 12 to 14 minutes, rotating the baking sheets if needed for even baking. Transfer the cookies to a rack to cool.
Make the filling: In a small bowl, cream the confectioners' sugar, butter, and peanut butter until smooth.
Add the heavy cream; continue creaming until smooth and fluffy.
Stir in the chopped peanuts and chocolate.
Assemble the sandwiches: Transfer the cooled cookies to a work surface, flipping half of them over.
Spread a scant teaspoon of filling onto each turned-over cookie.
Set another cookie on top of each filled cookie, pressing gently to spread the filling.
Store sealed at room temperature or in the refrigerator.