Position a rack in the middle of the oven and preheat the oven to 350 degrees F. Line a 13by-9-inch baking pan with foil, letting the excess hang over the edges of the pan by about 1 inch (this will help you lift the bar easily from the pan in one piece). Spray the pan with cooking spray.
Mix the flour, baking soda and salt in a medium bowl. In a large bowl, whisk the melted butter and sugars until combined. Use a whisk or a rubber spatula to mix in the egg, egg yolk, peanut butter and vanilla until smooth. Fold in the flour mixture until just combined (be careful not to over mix the dough or the bars will be tough). Fold in the chocolate chips and the peanuts.
Scrape the dough into the prepared pan, pressing and smoothing the top with a rubber spatula. Bake until the top is slightly golden and firm to the touch and the edges are starting to pull away from the sides of the pan, 30 to 35 minutes. Let cool completely on a wire rack.
Lift up on the foil overhang to remove the bar from the pan, and transfer it to a cutting board. Use a ruler to cut it into 2-inch squares.