Grease a 9×9-inch pan or line it with parchment paper and set aside.
In a medium saucepan over low heat, melt the butter. Stir in the packed brown sugar and vanilla extract until combined.
Add the quick cooking oats to the butter mixture and cook on low, stirring, for 2–3 minutes until the oats are evenly coated and slightly softened; remove from heat.
Press half of the oat mixture firmly into the prepared pan to form an even layer.
In a separate small saucepan over low heat (or in a microwave-safe bowl), melt 1 cup semisweet chocolate chips with 1/2 cup creamy peanut butter, stirring until smooth.
Pour the melted chocolate-peanut butter mixture over the oat layer in the pan and spread evenly.
Crumble the remaining oat mixture over the chocolate layer and press gently so it adheres but still leaves some texture.
Melt additional chocolate chips for the drizzle, then drizzle over the top of the bars.
Cover and refrigerate for a few hours (about 2 hours) until firm. Bring to room temperature briefly before cutting into bars and serve.