In a large bowl, or the bowl of a stand mixer, combine the butter and both sugars on a medium/high speed for about 3 minutes. Add in the egg, peanut butter and vanilla and combine again.
Add in the flour, baking powder and baking soda. Mix until all combined to a thick cookie dough. Chill in the fridge for 30 mins (or leave overnight).
Preheat the oven to 200C/180C Fan/Gas 6. Line 3 large baking trays.
Scoop the chilled cookie dough into balls (about 1 tbsp each) and place on the baking trays (2 inches apart, they spread a little). You should have approx 14 balls.Optional: Press with a fork to create the classic crisscross pattern Bake for 10-12 minutes, until golden (they will continue to cook a little after baking, but we want them firmer than a classic PB cookie). Let the cookies cool on a wire rack.
Whilst cooling, make the buttercream filling by first whipping the butter and jam together on a high speed with an electric/stand mixer.
Add half of the icing sugar, combine. Then repeat for the remaining icing sugar. The texture should be thick but spreadable (if too thin, add icing sugar. If too thick, add some milk).
Once the cookies are completely cool, spread 1 tbsp of the frosting onto the back of half the cookies. Place a small tsp dollop of the jam into the centre, then sandwich another cookie to it - to create cookie sandwiches!