Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder, and salt. Set the mixture aside.
In a large bowl, beat the softened butter, granulated sugar, and brown sugar together until creamy.
Add the vanilla extract, 1/2 cup of peanut butter, egg, and milk to the butter mixture and mix until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Scoop tablespoon-sized amounts of dough onto the prepared baking sheet. Flatten each scoop slightly, place a small spoonful of the remaining peanut butter in the center, and fold over the edges of the dough to seal.
Bake in the preheated oven for 8 to 10 minutes, or until lightly golden. Allow the cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Notes
For an extra crunch, consider adding chopped peanuts into the dough before baking.