Spray a square baking dish with nonstick cooking spray or line with parchment paper (leaving some paper hanging over the edges), set aside.
To make the brown butter, melt the stick of butter in a large saucepan or skillet over medium heat, stirring frequently.
Continue cooking for 5-8 minutes, stirring constantly, until some of the foam begins to reduce and the little bits at the bottom of the pan (milk solids) begin to brown.
Quickly remove the skillet from the heat and gently stir in the peanut butter, corn syrup, sugar, and vanilla extract.
Return the mixture to heat and reduce heat to low.
Continue cooking, stirring constantly, until the sugar is completely dissolved into the mixture, then remove from heat again.
Fold in the rice cereal using a silicone spatula or wooden spoon until all of the cereal is saturated with the peanut butter mixture.
Spread the cereal mixture into your prepared baking dish, press it into an even layer, and set aside to cool.
In a microwaveable bowl, melt together the chocolate and butterscotch chips in 20 second increments, stirring each time until combined and smooth.
Pour the chocolate/butterscotch mixture into the baking dish over the cereal mixture and spread into an even layer.
Allow the scotcheroos to cool until completely set (about one hour), cut into bars and serve.