Preheat the oven to 400 degrees. Line baking sheets with Silpats or parchment paper; set aside. In a small bowl, sift together flour, baking powder, and salt; set aside.
In a heatproof bowl, or in the top of a double boiler, set over a pot of simmering water, melt together unsweetened and semisweet chocolates and butter; set aside to cool.
In the bowl of an electric mixer fitted with the whisk attachment, beat eggs, sugar, and vanilla on medium speed until light and fluffy, about 3 minutes. Add cooled, melted chocolate mixture, and combine on low speed. Add flour mixture, and continue to beat on low speed until almost combined, about 1 minute. Remove bowl from mixer; stir in milk chocolate by hand.
Using a 1/2-ounce ice-cream scoop, drop dough onto prepared baking sheets about 3 inches apart. Bake until set, 12 to 15 minutes. Let cookies cool completely on baking sheets.