01 - Preheat oven to 350°F. Line two baking sheets with parchment paper.
02 - In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
03 - In a large bowl, beat butter and granulated sugar with an electric mixer until light and fluffy, about 2 to 3 minutes.
04 - Add the egg and vanilla extract to the butter mixture. Mix until fully combined.
05 - Gradually add the dry ingredients to the wet mixture, alternating with milk. Mix until just combined, avoiding overmixing.
06 - Scoop rounded tablespoons of dough onto prepared baking sheets, spacing cookies about 2 inches apart.
07 - Bake for 9 to 11 minutes, until edges are set and centers are barely golden. Let cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
08 - In a medium bowl, beat peanut butter and butter until smooth. Add powdered sugar, milk, vanilla extract, and salt. Beat until creamy and spreadable, adding more milk if needed for desired consistency.
09 - Once cookies are completely cool, frost each with a generous layer of peanut butter frosting. Decorate with sprinkles or chopped peanuts if desired.