Preheat the oven to 350°F and prepare two large baking sheets by lining them with parchment paper. In a separate small bowl, whisk the flour, baking soda, and salt until evenly combined. In the bowl of a stand mixer fitted with the paddle attachment or using a large mixing bowl with an electric hand mixer, blend the peanut butter, softened butter, brown sugar, and granulated sugar on medium speed for about 3 minutes until the mixture appears light and fluffy. Incorporate the eggs individually, then add the vanilla extract, stopping to scrape down the sides of the bowl as needed before mixing again until fully integrated, approximately 1 minute more. With the mixer set to low speed, gradually add the dry ingredients, mixing only until combined. Gently fold in the semisweet chocolate chips. Drop rounded tablespoons of dough, spacing them about 2 inches apart on the prepared baking sheets. If preferred, press the tops lightly with the tines of a fork to flatten. Bake one tray at a time for 10 to 15 minutes, or until the edges turn golden while the centers remain soft. Remove from the oven and allow the cookies to rest on the baking sheet for 1 to 2 minutes until slightly firmed, then transfer to a wire rack to cool completely. Store cookies in an airtight container for up to 3 days or freeze for longer storage.