Line an 8-inch or 9-inch square baking dish with parchment paper, leaving some overhang on the sides for easy removal. A 9x13-inch pan can also be used for thinner bars.
Add the butter and brown sugar to a large saucepan over medium heat. Cook until the butter melts and the sugar dissolves, stirring frequently.
Stir in the vanilla extract, then add the rolled oats and mix until evenly coated.
Reduce the heat to low and cook for 3–4 minutes, stirring occasionally, until the mixture is well blended.
Press half of the oat mixture firmly and evenly into the prepared pan.
Place the chocolate chips and peanut butter in a microwave-safe bowl. Heat in short intervals, stirring between each one, until completely smooth.
Pour most of the chocolate-peanut butter mixture over the oat layer, reserving about 1/4 cup for drizzling. Spread the filling evenly.
Add the remaining oat mixture over the chocolate layer. Press it down gently but firmly to create an even top layer.
Drizzle the reserved chocolate mixture over the top.
Refrigerate the no bake chocolate oatmeal bars for 2–3 hours, or overnight, until firm.
Let the bars sit at room temperature briefly before cutting into squares.