Preheat oven to 350°. Line two cookie sheets with parchment or silpat liners.
In a large mixing bowl, beat together the butter, brown sugar, baking powder, baking soda, salt, and vanilla until creamy. Add the egg and beat until smooth. Add the cocoa, mix to combine.
Add the flour to the batter alternately with the egg nog, beating until smooth. Scrape down the sides and bottom of the bowl, and beat again briefly.
Drop the dough by the ¼-cupful (I used a #16 ice cream scoop) onto the prepared baking sheets, leaving plenty of room between the cakes; they'll spread.
Bake the cakes in a preheated moderate oven for 15 to 16 minutes, until they're set and starting to firm up. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.
7 minute Icing
In the top of a double boiler, combine the sugar, water, egg whites, cream of tartar, and a pinch of salt. Beat with an electric mixer at low speed for 30 seconds, then set the pan over boiling water (the pan shouldn't touch the water).
Beating at high speed, cook the frosting for about 7 minutes, or until it's stiff and glossy. Remove it from the heat, add the vanilla, and beat an additional 2 minutes.