For the best texture, use regular, not natural-style, peanut butter.
Preheat oven to 350 degrees. Whisk together flour, baking soda, and salt in a medium bowl; set aside.
Put butter and sugars into the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed until pale and fluffy, 2 to 3 minutes. Mix in eggs. Add peanut butter; mix until combined.
Reduce speed to low. Add flour mixture in 3 batches, mixing well after each addition. Refrigerate dough 30 minutes.
Using a tablespoon or a 1/4-ounce ice cream scoop, drop dough onto baking sheets lined with parchment paper, spacing 2 inches apart. Press cookies with fork tines to flatten and create a cross-hatch pattern. Bake until golden brown, about 14 minutes. Let cool on sheets on wire racks 5 minutes. Transfer cookies to wire racks to cool completely. Cookies can be stored in airtight containers at room temperature up to 3 days.