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