Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large bowl, cream together the peanut butter, softened unsalted butter, granulated sugar, and light brown sugar until smooth and creamy using an electric mixer.
Add the egg and vanilla extract to the peanut butter mixture and beat until fully combined.
In a separate medium bowl, whisk together the all-purpose flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Stir in the old-fashioned rolled oats and semisweet chocolate chips until evenly distributed.
Using a tablespoon or cookie scoop, drop rounded spoonfuls of dough onto the prepared baking sheets, spacing them about 2 inches apart.
Flatten each cookie slightly with the back of a spoon or your fingers.
Bake in the preheated oven for 10 to 12 minutes, or until the edges are lightly golden but the centers are still soft.
Remove the cookies from the oven and allow them to cool on the baking sheets for 5 minutes.
Transfer the cookies to a wire rack to cool completely before serving.