Beat the butter, peanut butter, and sugar together using a stand or hand mixer, until light and fluffy. Add the eggs, one at a time, along with the vanilla extract, beating well between each addition.
Add the flour to the bowl, along with the baking powder and salt, and mix together on low until a dough forms. Split the dough in half, form into balls, and wrap in cling film or plastic wrap. Place in the fridge for an hour if possible, as this will make the dough much easier to work with, especially if you are going to stamp the top of the cookies. You can bake the dough without refrigerating, though, and it still shouldn’t spread.
When you are ready to roll out the dough, preheat the oven to 180 °C / 355 °F (160 °C fan), and line two large baking sheets with baking parchment or a silicone mat. Roll out the first ball of dough on a floured surface using a lightly floured rolling pin, until about ¼ inch thick. Using cookie cutters of your choice, cut out the cookies and place them on the baking sheets.
Bake the cookies for about 8–10 minutes, depending on the size of the cookies. Turn the baking sheet around halfway through baking to make sure they are evenly baked. If you prefer your cookies soft, like I do, then remove them from the oven as soon as they start to go ever so slightly brown at the edges. Leave the cookies to cool for 5–10 minutes on the baking sheet, and then transfer to a wire rack to finish cooling.