Adjust rack to lower third of oven and preheat oven to 350 degrees F. Sift the flour, baking soda, baking powder, and salt onto a sheet of waxed paper. Using an electric mixer on low speed, cream the butter, sugars, and peanut butter just until thoroughly blended. Mix in the eggs and vanilla until well combined. Add the dry ingredients, then the walnuts and chocolate chips. Using a rubber spatula, lightly stir in the rice cereal by hand so as not to crush the cereal into crumbs.

Drop rounded tablespoonfuls of batter onto parchment-lined baking sheets, about 2 inches apart, and bake for 9 to 11 minutes, or until cookies are golden. (Do not overbake.) Remove cookies from the parchment paper to a wire rack to cool.