Place 1/2 cup of butter, peanut butter, powdered sugar, and vanilla extract in a large bowl. Beat using an electric mixer until smooth. The dough will be very thick and a little crumbly.

Line a 9x13-inch pan with parchment paper. Transfer the peanut butter dough into the pan. Press the dough firmly into the dish.

Add chocolate chips and the remaining 1 tablespoon of butter into a microwave-safe bowl. Microwave in 30-second increments, stirring between intervals, until smooth (about a minute and a half total).

Pour the melted chocolate over the top of the bars and spread evenly. Let sit at room temperature until the chocolate has set, 30-60 minutes. Carefully remove the bars from the pan by pulling up on the excess parchment paper, and transfer to a cutting board. Slice with a sharp knife.

Place in the refrigerator for at least 1 hour, or until ready to serve.