Preheat oven to 375°F. Line a couple baking sheets with silicone baking mats or parchment paper and set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream together the butter and sugar for a couple minutes. Add in the egg and beat on high for 8 minutes, until fluffy and light. Add in the dry ingredients and beat on low until almost combined. Add in the Butterfingers on the lowest setting.
Scoop the dough into roughly 1 and 1/2 tablespoon-sized mounds onto your prepared baking sheets, an inch or two apart (I like using a medium-sized cookie scoop).
Bake for 10 minutes, until lightly golden. Allow them to cool for 5 minutes on the baking sheet before removing to a wire rack to cool further.
In the bowl of an electric mixer fitted with the paddle attachment beat together the butter, salt, and peanut butter, until combined. Add in half of the confectioners’ sugar on low, stir to combine, then add the other half and mix to combine. Turn the speed up to high and beat for 10-15 minutes until fluffy and glossy. Stir in the chopped Butterfinger pieces.
Scoop 1-2 tablespoon-sized mounds onto the bottom of one cookie, and top with another cookie (I used the same scoop I used to scoop the cookies for the frosting).