Heat oven to 350° F with the racks in the upper and lower third. Line 2 baking sheets with parchment. In a medium bowl, whisk together the flour, baking powder, and salt; set aside.
In a separate bowl, beat the butter, peanut butter, granulated sugar, and brown sugar with an electric mixer on medium-high speed until evenly combines, 2 to 3 minutes. Beat in the eggs and vanilla. Reduce speed to low and gradually add the flour mixture, mixing just until combined (do not overmix). Fold in the bananas and 1 cup of the peanuts (the dough will be very stiff).
Drop heaping tablespoonfuls of the dough onto the prepared baking sheets, spacing them 2 inches apart. Sprinkle with the remaining 1/2 cup of peanuts. Bake, rotating the sheets halfway through, until the cookies are golden, 15 to 18 minutes. Cool slightly on the baking sheets, then transfer to racks to cool completely.
Storage suggestion: Keep the cookies in an airtight container at room temperature for up to 5 days.