Heat oven to 350° F. Line 2 baking sheets with parchment. In a large bowl, mix together the peanut butter, sugars, eggs, baking soda, and salt until smooth.
Shape the dough into balls (about 1 heaping tablespoon each) and place 2 inches apart on the prepared baking sheets. Using the tines of a fork, flatten each ball, creating a crisscross pattern.
Bake, rotating the baking sheets halfway through, until puffed and golden, 10 to 12 minutes (the cookies will deflate as they cool). Let cool slightly on the baking sheets, then transfer to a wire rack to cool completely.