Heat oven to 350°F Line several baking sheets with parchment paper; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter, peanut butter, and both sugars together until light and fluffy. Add egg and vanilla, and mix on medium speed until well combined.
In a medium mixing bowl, sift flour and baking soda together. Add to the butter mixture, and beat just to combine.
Scoop out 2 tablespoons of dough, shape into a ball, and place on one of the prepared baking sheets. Repeat with remaining dough, placing scoops 3 inches apart. Dip the tines of a fork in warm water; press the dough balls lightly with the back of the fork to flatten them slightly (dip the fork back in the water for each dough ball, to prevent sticking). Sprinkle with a few peanuts, if desired.
Transfer to the oven, and bake until golden brown, 18 to 20 minutes, rotating the baking sheets between the oven shelves halfway through baking. Transfer baking sheets to a wire rack to cool. Store in an airtight container at room temperature up to 1 week.
Makes 2 dozen.