Line the 8-inch square pan with parchment paper, leaving some overhang for easy removal.
In a microwave-safe bowl, melt the butter and peanut butter together for about 30 seconds. Stir until the mixture is smooth, glossy, and fragrant with a nutty aroma.
Add the sugar and vanilla extract to the melted mixture. Stir well until the sugar dissolves and the mixture turns creamy and slightly thickened.
Stir in the graham cracker crumbs until evenly coated; the mixture will become thick and sticky.
Press the peanut butter mixture evenly into the prepared pan, using the back of a spatula or your fingers to create a smooth, compact surface.
Chop the chocolate chips finely to ensure smooth melting and a glossy finish.
In a microwave or double boiler, melt the chocolate chips until smooth and shiny, about 1-2 minutes, stirring frequently.
Pour the melted chocolate over the pressed peanut butter layer, spreading it out evenly with a spatula for a glossy, smooth top.
Refrigerate the bars for at least 2 hours until the chocolate is firm and the bars are set.
Once chilled and firm, use the parchment overhang to lift the bars from the pan. Cut into squares and enjoy your homemade no-bake peanut butter bars!