Preheat the oven to 350ºF (180ºC). Line two large baking sheets with parchment paper.
In a medium-sized bowl, whisk together the all purpose flour, cake flour, baking soda and salt until well blended.
In a large bowl, beat the peanut butter, butter and shortening, with an electric mixer on low speed, until smooth. Beat in the egg and vanilla, on medium speed, until well combined.
Set aside 1/4 cup (50g) of the granulated sugar. Add the remaining granulate sugar and the brown sugar to the peanut butter mixture. Beat on medium speed, until light and fluffy. Add the flour mixture and beat on low speed, 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.