This version of the childhood combination concentrates the flavors into a sweet dessert that appeals to all ages.
Heat oven to 350 degrees. Grease a 9-by-13-inch pan with butter, and line the bottom with parchment paper. Grease the parchment, and coat inside of pan with flour; set aside. Place butter and sugar in the bowl of an electric mixer fitted with the paddle attachment. Beat on medium-high speed until fluffy, about 2 minutes. On medium speed, add eggs and peanut butter; beat until combined, about 2 minutes.
Whisk together salt, baking powder, and flour. Add to bowl of mixer on low speed; combine. Add vanilla. Transfer two-thirds of mixture to prepared pan; spread evenly with offset spatula. Using offset spatula, spread jam on top of peanut-butter mixture. Dollop remaining third of peanut-butter mixture on top of jam. Sprinkle with peanuts.
Bake until golden, about 45 minutes. Transfer to a wire rack to cool; cut into about 36 1 1/2-by-2-inch pieces.