Preheat the oven to 350° and position 2 racks in the lower third. In a medium bowl, whisk together the flour, baking soda, baking powder and salt. In a large bowl, using an electric mixer, beat the butter with the brown sugar and granulated sugar until fluffy. Add the eggs and vanilla and beat at low speed until blended. Add the peanut butter and beat until combined. Beat in the dry ingredients just until incorporated. Beat in the chocolate chunks and peanuts.
For each cookie, use an ice cream scoop or a large spoon to scoop 3 tablespoons of the dough onto parchment-lined or nonstick baking sheets; space them 2 inches apart. Bake for about 13 minutes for slightly chewy cookies and 15 minutes for crisp cookies; shifting the pans from top to bottom and front to back for even baking. Let the cookies cool for 10 minutes on the baking sheets, then transfer them to a wire rack to cool completely.
The cookies can be stored in an airtight container for up to 1 week.