Preheat the oven to 350°F (175°C). Line an 8x8-inch pan with parchment paper, leaving overhang for easy removal, and lightly grease it.
Whisk together melted butter, granulated sugar, and brown sugar. Stir in the beaten egg and vanilla. Add flour, baking soda, and salt. Mix until combined. Press 2/3 of the dough into the pan and set aside the rest for the topping.
Mix peanut butter, softened butter, and powdered sugar until smooth. Spread evenly over the cookie base, leaving a border.
Crumble the reserved dough over the peanut butter layer to create a topping.
Bake for 25–30 minutes until the edges are golden brown and the center is set. Let cool completely in the pan.
Microwave chocolate chips and butter in 30-second intervals until smooth. Spread over the cooled bars and let set for 15–20 minutes.
Lift the bars from the pan using the parchment overhang. Slice into squares or rectangles and serve.