To make the cake, position a rack in the center of the oven and preheat to 350ºF (175ºC). Rub three 8-inch round cake pans with softened butter, line each with a round of parchment paper, and butter the parchment. Dust the pans with cocoa powder, tapping out the excess.
Trim the zucchini and grate them on the medium holes of a box grater (the ones that measure 1⁄8 inch [2–3 mm]); you should have 2 cups packed grated zucchini. In a large bowl, whisk together the brown sugar, eggs, oil, milk and vanilla extract, until combined well. In a medium bowl, sift together the cocoa powder, sorghum flour, sweet rice flour, baking powder and salt, pushing through any clumps. Stir half of the flour mixture into the egg mixture until smooth, then add the grated zucchini, stirring until smooth, followed by the remaining flour mixture.
Divide the batter among the prepared pans and smooth into an even layer. Bake the cakes until the tops spring back to the touch, and a toothpick inserted near the center comes out with moist crumbs, 30-40 minutes. Let cool completely in the pans, at least 1 hour, then turn out and peel away the parchment. (The cakes can be wrapped and stored for up to 1 day.)
To make the frosting, combine the cream cheese, butter, powdered sugar, matcha powder and salt in the bowl of a stand mixer fitted with the paddle attachment. Beat the frosting on low speed, increasing to medium speed until smooth, light and fluffy, 2–3 minutes, scraping down the sides of the bowl occasionally.
To assemble the cake, place one cake layer on a serving board or platter and use a small, offset spatula or butter knife to spread about one-fourth of the frosting over the top in an even layer. Top with a second cake layer and repeat, then repeat with the remaining cake layer, frosting the top and sides. If you like, decorate the top of the cake with chocolate discs or shavings, a sprinkle of matcha, and a touch of edible gold leaf (if you’re weird like me and happen to have it lying around).
For the cleanest results, chill the cake to firm up the frosting, at least 1 hour. Cut the cake with a large, sharp chef’s knife dipped in hot water and wiped dry between cuts. The cake is best within a day or two of baking, stored refrigerated airtight.