Line an 8-inch square baking pan with parchment paper, leaving extra overhang on the sides for easy removal.
In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma.
Once browned and nutty, remove from the heat and stir in the brown sugar. Add the peanut butter and vanilla, stirring until combined. Gradually stir in the powdered sugar, a cup at a time, until mixture is smooth. Pour mixture into prepared pan and place in the freezer until firm, about 30 minutes.
Place the chocolate chips and butter in a microwave-safe bowl and heat in the microwave for 60-90 seconds.** Remove and stir until smooth. Spread evenly over chilled peanut butter filling. Refrigerate until firm and set, at least one hour.
Use a knife to loosen the edges before using the parchment sides to lift the bars out of the pan. Cut into squares and serve.