Heat the oven to 350°F (175°C). Lightly grease a 9 x 13-inch (23 x 33 cm) metal baking pan and line the bottom with parchment paper, leaving enough overhang on two sides to help lift the cooled cake.

In a medium bowl, whisk together the flour, baking powder, salt, 1 teaspoon cinnamon, and nutmeg. Whisk thoroughly so the leavening and spices are evenly distributed.

In a large bowl, beat the eggs, granulated sugar, and ½ cup brown sugar for about 1 minute, until well combined and slightly lighter in color. Add the mashed sweet potato, vegetable oil, sour cream, and vanilla. Mix until smooth and uniform. Small traces of sweet potato are fine, but there should be no large lumps.

Add the dry mixture to the sweet potato mixture. Fold with a rubber spatula just until no dry streaks remain. The finished batter should be thick but easily spreadable. Avoid prolonged mixing once the flour is incorporated.

Stir together the ¾ cup brown sugar, 1 tablespoon cinnamon, and melted butter. The mixture should resemble damp, loose brown sugar rather than a liquid sauce.

Spread approximately half of the cake batter evenly across the prepared pan. Because the batter is fairly thick, use a spatula to gently push it into the corners without disturbing the parchment.

Sprinkle the cinnamon-brown sugar mixture evenly over the batter, breaking up any large clumps with your fingers. Leave a narrow border around the edges to reduce the chance of the sugar sticking to the sides of the pan.

Spoon the remaining batter across the cinnamon layer in small portions, then carefully spread it into an even layer. Use a light touch so the filling remains mostly in the center instead of mixing into the batter.

Bake at 350°F (175°C) for 35 to 40 minutes, or until the top is set and lightly browned and a wooden pick inserted into a cake-only area near the center comes out with a few moist crumbs rather than wet batter. If the pick passes directly through the sugar ribbon, it may appear wetter than the cake itself.

Set the pan on a wire rack and let the cake cool completely, about 1 hour. Do not add the icing while the cake is warm; the butter-based shreds will melt.

Beat the softened butter until creamy: Gradually beat in the powdered sugar, then add the vanilla, milk, and salt. Continue mixing until a very thick, smooth icing forms. It should be substantially firmer than a pourable cake glaze.

Form the icing into a compact block about 1 inch (2.5 cm) thick, wrap it in parchment paper, and refrigerate for 30 minutes, or until firm enough to grate without smearing.

Unwrap the chilled icing and grate it over the completely cooled cake using the large holes of a box grater. Work evenly across the surface. If the icing begins to soften in your hands, refrigerate it briefly before continuing.

Cut the cake into 15 pieces: For the cleanest presentation, wipe the knife between cuts and avoid pressing heavily on the grated icing.