Step 1In the bowl of a stand mixer using the paddle attachment, or in a large bowl using a hand mixer, cream the butter over medium speed just until creamy. Add the sugars to the butter and cream together, careful not to over-mix. Scrape down the bowl to make sure the sugars and butter are evenly combined.
Step 2Beat in the egg just until incorporated, then beat in the peanut butter, scraping down the bowl again after mixing.
Step 3In a medium bowl, whisk together the flour, baking soda and salt.
Step 4Add the flour mixture to the mixing bowl, beating until smooth and the flour is evenly incorporated. Scrape down the bowl, making sure the ingredients are combined and smooth.
Step 5Remove the dough from the bowl and form it into a rough log, approximately 2 1/2 inches thick. Place the dough in a sheet of parchment paper and roll the dough in the paper so it is smooth and a consistent 2 1/2-inches thick throughout the log. The log will be about 9 inches long.
Step 6Place the peanuts in a rimmed cookie sheet and gently roll the dough in the peanuts to coat on all sides. The peanuts should stick to the log; if they don't, brush the roll with a beaten egg, then roll it in the peanuts so they stick. Wrap the finished log in parchment paper or plastic wrap and refrigerate until well-chilled, at least 2 hours, up to overnight.
Step 7Heat the oven to 350 degrees. Slice the cookies into 1-inch-thick slices and place on parchment-lined baking sheets, spacing them at least 3 inches apart, as the cookies will spread while they bake. Bake the cookies until set and browned around the edges, about 20 minutes, rotating the baking sheet halfway through baking for even coloring. Cool the cookies, still on the baking sheet, on a rack until set and cool enough to handle.