Preheat the oven to 350 degrees Fahrenheit and lightly grease a half-sized baking sheet pan (18"x13").
In the bowl of an electric mixer, cream together butter and sugars. Add eggs, peanut butter, and vanilla and mix until incorporated. Add in baking soda, salt, flour and oatmeal and mix just until incorporated.
Press dough evenly into prepared pan and bake for 12-15 min. Check it at 10 minutes and every minute after that until they're set, so they don't over bake.
Allow them to cool to room temperature, then melt the white chocolate chips and spread over. I melted mine in the microwave in 30 second increments, stirring in between.
Place bars in the fridge for a few minutes so the white chocolate can quickly set up. Then melt the semi-sweet chocolate chips and spread over the white chocolate layer that has set. Allow the chocolate to set completely.
Slice into squares and serve.