In an electric mixer, fitted with the paddle attachment, or with a hand-held electric mixer, combine the butter and brown sugar on low speed until just incorporated. Raise the speed to high and mix until light and fluffy, about 10 minutes. (Occasionally turn the mixer off, and scrape the sides of the bowl down with a rubber spatula.)

Add the egg and peanut butter and continue mixing until fully incorporated.

Meanwhile, in a small bowl, whisk together the flour, baking soda, and salt.

Reduce the speed of the mixer to low and add the flour mixture. Raise the speed to medium and mix briefly until a smooth dough is formed. Form the dough into a disc, wrap it in plastic wrap, and refrigerate until firm.

Preheat the oven to 375 degrees F. Line 2 baking sheet pans with parchment paper. Using a 2-teaspoon-sized ice cream scoop, melon baller, or your hands, form the dough into about 50 small balls. Put sugar on a pie plate or dish. Roll the dough balls in the sugar to coat lightly. Arrange the dough about an inch apart on the prepared pans.

Using the tines of a fork, flatten the dough balls slightly, while making cross-hatch marks on their tops. Bake the cookies in batches, turning the pans once, until lightly browned, about 15 minutes per batch. Let the cookies cool slightly on the baking sheets, and then transfer to racks to cool completely. Store in a sealed container.