In a bowl, stir together flour, baking soda, baking powder, and salt.
Beat together the butter, peanut butter, and sugars until light and fluffy. Beat in eggs and vanilla. Gradually beat in the flour mixture. Stir in chopped peanuts, if using.
Wrap dough in plastic wrap and refrigerate until chilled, or about 2 to 4 hours.
Take tablespoonfuls of dough and roll into balls with hands. Place about 3 inches apart on greased baking sheets. Using a fork dipped in flour, lightly press cookies, flattening and forming a criss-cross pattern on each cookie with the tines.
Bake at 375° until golden brown, or about 10 to 12 minutes. Cool on baking sheets for a minute; transfer to rack to cool completely.
Makes about 4 dozen peanut butter cookies.