1. Position racks in upper and lower thirds of oven and preheat to 350°F. Butter 2 large baking sheets. Whisk together flour, baking soda and salt in a medium bowl.
2. With an electric mixer on medium-high speed, beat butter with both sugars for about 3 minutes, until light and fluffy. Reduce speed to medium and add eggs and vanilla; beat until smooth and well combined. Beat in peanut butter. Scrape down sides of bowl. Reduce speed to low and beat in flour mixture in 2 batches, until just combined.
3. Form dough into 1-inch balls and place on baking sheets. Flatten each ball by pressing with tines of a fork in a crisscross pattern. Bake, switching pans halfway through, 14 to 16 minutes, or until golden brown on bottom and barely colored on top. Cool on pans on wire racks for 5 minutes. With a spatula, carefully transfer cookies to a wire rack and allow to cool completely. Repeat with remaining cookie dough.
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