Preheat oven to 350 degrees F and line baking sheets with parchment paper.
In a large bowl with an electric mixer, beat butter and sugars until smooth and creamy. Add eggs and vanilla and beat until combined.
Add oats, flour and cocoa and mix until combined. Stir in white chocolate chips and chocolate chips. Dough will be slightly tacky still.
Drop rounded tablespoonfuls onto prepared baking sheets and press down slightly. Bake for 8-10 minutes, until just slightly glossy in the center (a tiny bit!) but mostly set. Let cool for 10 minutes before removing to wire racks to cool completely.