Preheat the oven to 350 degrees F. Cream the butter and both sugars until light and fluffy in the bowl of an electric mixer fitted with the paddle attachment. Add the vanilla, then the eggs, 1 at a time, and mix well. Add the cocoa and mix again. Sift together the flour, baking powder, baking soda and salt and add to the chocolate with the mixer on low speed until just combined. Fold in the chopped white chocolate. Drop the dough on a baking sheet lined with parchment paper, using a 1 3/4-inch ice cream scoop or a rounded tablespoon. Dampen your hands and flatten the dough slightly. Bake for exactly 15 minutes (the cookies will seem underdone). Remove from the oven and let cool slightly on the pan, then transfer to a wire rack to cool completely.