For the cakes: Preheat the oven to 350 degrees F, or 325 degrees F on the convection setting. In a medium bowl, whisk together the flour, cocoa powder, baking soda and salt. In a large bowl, using an electric mixer on medium speed, beat the butter and sugar until pale and fluffy, about 3 minutes. Add the egg and beat to combine. Beat in the flour mixture in three additions, alternating with additions of the buttermilk, starting and ending with flour mixture.
Scoop the batter by 1/4-cupfuls onto three parchment-lined rimmed baking sheets, at least 3 inches apart. With the back of a cookie scoop or a spoon, shape the scoops of batter into neat circles. Bake until the cakes spring back when gently pressed in the center, 15 to 18 minutes. Slip the parchment with the cakes still on it onto cooling racks. Let the cakes cool completely on the parchment.
For the filling: In a medium bowl, beat the peanut butter and butter with an electric mixer on medium speed until smooth and creamy. Add the confectioners' sugar and the salt and beat on low until smooth. Chill the filling for 15 minutes.
Carefully lift the cakes off of the parchment. Spread about 3 tablespoons of filling on a flat side of half of the cakes. Top with the remaining cakes, flat-side down.