Preheat oven to 350°F (180°C). In a small bowl, whisk flour with baking soda and salt until mixed. Using an electric mixer, in a large bowl, beat peanut butter with butter and sugars until well blended. Beat in egg, scraping down sides to mix well. Beat in vanilla. Gradually beat in flour mixture. Stir in peanuts, if using, until evenly distributed.
Using floured hands, roll dough into 1-inch (2.5 cm) balls. Place at least 2 inches (5 cm) apart on ungreased baking sheets. Dip the flat base of a glass in flour and gently press on each ball to flatten cookie to a little thicker than 1/4-inch (0.5 cm). Then, dip the tines of a fork in flour and press gently into centre of cookie. Press again at a different angle to form a cross pattern.
Bake in centre of preheated oven until golden and just set, 9 to 11 minutes. Remove from oven. Leave on baking sheet for 1 minute, then transfer to a rack to cool completely.