In a medium mixing bowl, combine all purpose flour, bread flour, baking soda, baking powder and salt. Whisk well to thoroughly combine. Set aside.
In a large mixing bowl, using a handheld electric mixer, beat peanut butter and butter together until smooth.
Add sugars, and beat to combine. Scrape down sides and bottom of bowl with a rubber spatula.
Add eggs and vanilla to the bowl, and beat to combine.
Add flour mixture to wet ingredients. Beat on low speed until most of the flour is incorporated, then beat on medium speed until the mixture is uniform, about 3 - 4 minutes. Using a rubber spatula, fold cookie dough, making sure to scrape the bottom of the bowl, to ensure everything is well combined.
Use a small cookie scoop to portion roughly 6 dozen balls of cookie dough onto a parchment-lined baking sheet. Roll cookie dough in your hands to form a round ball. Dip the tops of the cookie dough in granulated sugar.
Return peanut butter cookie balls to baking sheet, cover tightly with plastic wrap and refrigerate for 30 minutes.
Preheat oven to 325°F with convection fan // 350° F without fan // 180°C // Gas mark 4.
Place cookie dough on parchment-lined baking sheets, leaving 2 inches between each one, and bake for 14 minutes. (Bake one sheet at a time for best results. Keep cookie dough in the fridge until ready to bake).
Remove from oven and let cool on baking sheet for 10 minutes before moving to cooling rack to cool completely. While cookies are cooling, make chocolate buttercream (recipe below).
When cookies have cooled completely, line them up in pairs, flipping over the bottom of the sandwich cookie.
Using a piping bag, or a small spatula, fill cookies with about 1 tablespoon of buttercream.
Place top cookie on buttercream and lightly press to secure in place.
Store in an airtight container at room temperature for 3 days, or refrigerate for up to 7 days.
Chocolate buttercream
Combine all ingredients for chocolate buttercream in a medium mixing bowl and beat on medium/medium-high speed for 3 - 4 minutes, until light and fluffy.
Notes
You can make the buttercream and the dough ahead of time and freeze for later. Store in an airtight container in the freezer for up to 3 months.