Lightly grease a 9 x 9-inch square pan or line it with parchment paper, leaving an overhang on two sides. Melt the butter in a saucepan over medium heat, then stir in the brown sugar and vanilla extract.

Add the quick-cooking oats to the butter mixture and cook, stirring constantly, for about 2 to 5 minutes. Continue until the oats are softened and the mixture is evenly blended, but do not boil it for an extended time.

Press approximately half of the warm oat mixture firmly into the prepared pan. Use the back of a spoon or a flat measuring cup to create an even, compact base.

Melt the semisweet chocolate chips and creamy peanut butter together in the microwave using short intervals, stirring between each interval, or melt them gently in a double boiler. Stir until glossy and completely smooth.

Spread the chocolate-peanut butter mixture evenly over the oat base. Work carefully toward the corners so the center and edges have a consistent layer.

Crumble the remaining oat mixture over the chocolate layer and gently press it into place. Avoid pressing so hard that the chocolate layer squeezes out around the edges.

Refrigerate the pan for 2 to 3 hours, or overnight, until the bars are firm. Let the chilled slab stand briefly at room temperature before lifting it from the pan and cutting it into about 16 bars.