In a small pot add the butter, and the unsweetened and semisweet chocolates and heat over low heat, stirring frequently, just until everything is melted and smooth. Allow to cool slightly.
Meanwhile, in a small bowl combine the flour, baking powder, and salt. In a large bowl beat the eggs and sugar with an electric mixer until creamy and yellow, about 1 minute. Beat in the vanilla. With the mixer on low speed, blend in the flour mixture in a few batches, just until barely combined. Add the chocolate mixture to the bowl and beat until smooth. Let the mixture cool; the batter will be very thick. Stir in the white chocolate chips and the dried cherries (if you stir the chocolate chips in before it cools the chocolate will start to melt and become streaky, which is also pretty but be aware that will happen).
Scoop out heaping tablespoons of the batter and create 12 mounds on a rimmed baking sheet (you will bake three baking sheets of cookies in all). Bake for 9 to 11 minutes until the tops are crackly and the edges are set. Remove from the oven and let them sit for one minute on the baking sheet (the cookies will sink slightly), then transfer them to a wire rack.