Position oven racks to the top and bottom thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment.
Beat the peanut butter, granulated and brown sugars and salt in a large bowl, with an electric mixer, on medium-high speed until light and fluffy, about 5 minutes. Beat in the vanilla and egg.
Put some granulated sugar in a small bowl. Roll tablespoons of the dough into balls and coat each in sugar. Evenly space them on the prepared baking sheets. Press each ball down with a fork, making a crisscross pattern, to flatten.
Bake until the bottoms are golden brown, 10 to 12 minutes. Let the cookies cool for a few minutes on the baking sheets, then transfer to a rack to cool completely. The cookies can be stored in an airtight container at room temperature for up to 3 days.
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