For the best texture, use regular, not natural, peanut butter.
Preheat oven to 350 degrees. Whisk flour, baking soda, and salt in a medium bowl.
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, then peanut butter.
Reduce speed to low. Add flour mixture in 3 batches, mixing until just combined after each addition. Cover with plastic wrap, and refrigerate 30 minutes.
Using a 1 1/2-inch ice cream scoop, drop dough onto baking sheets lined with parchment paper, spacing about 1 1/2 inches apart. Press cookies with fork tines to flatten and create a cross-hatch pattern. Bake until golden brown, 14 to 16 minutes. Let cool on sheets 5 minutes. Transfer cookies to wire racks to cool completely.
The cookies can be stored in airtight containers at room temperature up to 3 days.