Preheat oven to 350ºF (180ºC). Line two large baking sheets with parchment paper.
In a medium bowl, whisk together the all purpose flour, cake flour, baking soda and salt. Set aside.
In a large bowl, beat the peanut butter, butter and shortening until smooth.
Beat in the egg and vanilla until well combined.
Set aside 1/4 cup (50g) of the granulated sugar. Beat the peanut butter mixture with brown sugar and remaining granulated sugar, with an electric mixer at medium speed, until light and fluffy.
Add the flour mixture and beat just until combined.
Roll the dough into 1 1/2-inch (3.8cm) diameter balls (about 1 1/4 oz or 35g each). Roll the dough balls in the reserved granulated sugar.
Place the dough balls 3 inches apart onto the prepared baking sheets. Flatten with a fork, making a criss cross pattern.
Bake until the cookies are puffy and start to crack around the edges, about 9-11 minutes.
Let the cookies cool on baking sheet for 1 minute, then carefully transfer them to a cooling rack. The cookies will be fragile, but they will firm up as they cool.
Place a spoonful of filling on the bottoms of half of the cooled cookies. Top with remaining the remaining cookies.
Peanut Butter Filling
In medium bowl, beat the peanut butter and butter together until smooth.
Beat in the powdered sugar, heavy cream, vanilla and salt until smooth.