Preheat the oven to 300 degrees and position 2 oven racks in the middle and upper third of the oven. Line 2 baking sheets with heavy aluminum foil.

Sift the flour, cocoa, baking powder, and salt together. Set aside. Put the butter in the bowl of an electric mixer and mix on low speed for 15 seconds. Add the powdered sugar. Increase the speed to medium and beat until the mixture is smooth, about 1 minute. Decrease the speed to low and add the flour mixture. Mix until the flour mixture is blended completely into the dough and the dough holds together, about 2 minutes. Stop the mixer and scrape the bowl during this time. Mix in the chocolate chips. Divide the dough in half and place 1 piece of dough on waxed paper. Press the dough into a rectangle measuring 7 by 5-inches and about 1/2-inch thick. Use a large sharp knife to cut 8 strips 7-inches long by about 5/8-inch wide. Cut each strip into 3 cookies about 2 1/4-inches long. Using a wide spatula to move the cookies, place the cookies 1-inch apart on the prepared baking sheet. Repeat with the remaining dough.

Bake 25 minutes. The tops of the cookies change from shiny to dull. Cool the shortbread on the baking sheets for 10 minutes. Transfer the cookies to 2 wire racks to cool completely.

While the cookies cool, melt the white chocolate. Preheat the oven to 175 degrees. Place the white chocolate in a small nonreactive container. Melt the white chocolate in the oven, about 8 minutes. Remove the white chocolate from the oven as soon as it is melted and stir until smooth. Hold a cookie over the white chocolate and spoon white chocolate over 1 end of the cookie, covering about 1/2-inch of the cookie. Drag the bottom of the cookie over the top of a bowl to remove any excess chocolate. Let the cookies sit on the rack until the white chocolate is firm. To firm the white chocolate quickly, refrigerate or freeze the cookies.

Freezing instructions: Place 2 cookies side by side and wrap in plastic wrap. Put the wrapped cookies in a metal or plastic freezer container and cover tightly. Freeze up to 3 months. To defrost, remove as many cookies from the freezer as needed and defrost the wrapped cookies at room temperature. Serve within 3 days.