Preheat oven to 350ºF. Line 2 baking sheets with parchment. In a heavy medium saucepan over low heat, melt 2 Tbsp. butter with semisweet chocolate, stirring until smooth. Transfer to a bowl and let cool slightly.
Sift flour, cocoa, baking powder and salt onto a sheet of parchment. Using an electric mixer on medium-high speed, beat remaining 8 Tbsp. butter with sugar until light, about 3 minutes. Reduce speed to medium-low and add eggs 1 at a time, scraping down sides of bowl as needed. (Batter will look curdled.) Beat in melted chocolate mixture and vanilla until smooth. On lowest mixer speed, stir in flour mixture and then milk chocolate chips.
Drop by tablespoons 2 inches apart on baking sheets and bake until edges are firm, 10 to 12 minutes, rotating sheets front to back and top to bottom halfway through. Let cool on sheets on wire racks for 5 minutes, then remove cookies to racks to cool completely.
In a heatproof bowl set over a saucepan of simmering water (do not let water boil), melt white chocolate chips with shortening, stirring until smooth. Remove bowl from pan and let mixture cool slightly. Dip half of each cookie in white chocolate mixture, letting excess drip off. Place on waxed paper or parchment and let stand until chocolate has set.