Preheat oven to 350 degrees. Spray an 8 inch square pan (2 inches deep) generously with non-stick cooking spray. (I used an 8 inch round cake pan, 2 inches deep. As you can see in the photo, it baked right to the top. Probably a 9 inch round cake pan would have been better.)
In a medium bowl, whisk together flour, cocoa powder, baking soda and salt.
In the bowl of a stand mixer combine butter and brown sugar. Beat until pale and fluffy.
Add eggs, one at a time, beating well in between. Add vanilla.
Add flour mixture and 1 1/3 cups water, starting with some of the flour mixture, then adding some water alternately. End with the flour mixture and mix just until combined. Do not over mix.
Pour batter into prepared cake pan. Smooth top and bake for 45-55 minutes until tester comes out clean. In the 8 inch round pan mine took 54 minutes.
Cool in pan on rack for one hour. Transfer to cake plate to frost.
Frosting:
Combine all ingredients with 2 Tbsp. water in a metal bowl over simmering water. Beat with a handheld mixer for 6-7 minutes, until thick and fluffy.
Remove from water and continue to beat for 1-2 more minutes until somewhat cooled.
Mound the frosting on top of the cake, then sprinkle with cocoa powder.
The egg whites are not fully cooked, so you can substitute pasteurized or reconstituted dried egg whites.
I followed the recipe and made the cake the night before because they said it would improve in flavor if made 1 day ahead. I then wrapped it in plastic wrap and kept it at room temperature. I then unwrapped it and frosted it the next day.