Preheat the Oven: Preheat the oven to 350°F (175°C).
Combine Dry Ingredients: Sift the all-purpose flour and baking powder together, then whisk them thoroughly to distribute the baking powder evenly.
Cream Butter and Sugars: Add the room-temperature butter, brown sugar, and granulated sugar to a stand mixer fitted with a paddle attachment and cream until combined. If the peanut butter is not particularly salty, add ¼–½ teaspoon sea salt at this stage. Blend in Peanut Butter: Add the peanut butter to the creamed butter and sugar mixture, then mix until fully incorporated.
Add Egg and Vanilla: Mix in the room-temperature egg and vanilla extract until combined.
Finish the Dough: Add the flour and baking powder mixture to the wet ingredients and beat just until the flour mixture is incorporated.
Shape the Cookies: Roll the dough into approximately 1-inch balls and arrange them on a baking sheet lined with parchment paper.
Create the Criss-Cross Pattern: Press each dough ball gently with a fork, making a classic criss-cross pattern across the surface.
Bake the Cookies: Bake at 350°F (175°C) for about 10 minutes, removing them when they are almost done.
Cool and Set: Leave the cookies on the baking sheet to cool completely. They will be very soft immediately after baking, but the residual heat will allow them to continue setting and firm up as they cool. Do not transfer them until they are sufficiently set to handle.