Place ½ 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. Refrigerate for 1 hour. 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.
Return to the refrigerator for at least 1 more hour, or until ready to serve.
Storage: Refrigerate in an airtight container for up to 1 month. Freeze in individual portions for up to 4 months. Thaw in the refrigerator for 24 hours before serving.Cutting: For best results, refrigerate the bars for 30-60 minutes before cutting. Do not allow the chocolate to fully set before you cut it, or it will crack.