Add the melted butter, peanut butter, graham cracker crumbs, powdered sugar, brown sugar, and vanilla to a large mixing bowl.
Stir until everything is thoroughly combined and you have a thick, moist mixture that is easy to spread.
Transfer the peanut butter mixture to a 9×13-inch pan. Press and smooth it into an even layer with a spatula.
Melt the chocolate chips using a double boiler or microwave. If using the microwave, heat at 50% power in short 30-second intervals, stirring well after each interval until smooth.
Pour the melted chocolate over the peanut butter layer. Gently spread it from edge to edge with a spatula.
Lightly jiggle or tap the pan to help level the chocolate and smooth the surface.
Refrigerate the bars for about 1 hour, or until the peanut butter layer and chocolate topping are firm.
Allow the pan to sit at room temperature for a few minutes before slicing into bars. Serve chilled or slightly softened.