MIX flour, baking soda and salt in medium bowl; set aside.
BEAT butter, peanut butter and sugars in large bowl with electric mixer on medium speed until light and fluffy. Add egg and vanilla; beat well. Gradually beat in flour mixture. Stir in chocolate chunks and peanuts. Drop by heaping tablespoonfuls onto ungreased cookie sheets.
BAKE at 375°F for 11 to 13 minutes or just until golden brown. Cool on cookie sheets 1 minute. Remove to wire racks and cool completely.
Make-Ahead: After cookies are completely cooled, wrap in plastic wrap and place in an airtight plastic container or freezer zipper-style plastic bag. Cookies can be frozen for up to 1 month. Bring cookies to room temperature before serving.