Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking soda, salt, and baking powder. Set aside.
In a large mixing bowl, beat together the softened butter, smooth peanut butter, light brown sugar, and granulated sugar until the mixture is light and fluffy.
Add the egg and vanilla extract to the peanut butter mixture, mixing until well combined.
Gradually add the dry ingredients to the wet mixture, mixing until just combined. Be careful not to overmix.
Fold in the old-fashioned oats, giving the cookie dough a hearty texture.
Using a cookie scoop or tablespoon, drop rounded balls of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake in the preheated oven for about 10-12 minutes, or until the edges are lightly golden. The centers may appear slightly underbaked; they will firm up as they cool.
Allow the cookies to cool on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.