Step 1

Preheat the oven to 375ºF. Line the bottom of a large (about 12- by 15-inch) rimmed baking sheet with parchment paper.

Step 2

Make the cake: Separate the egg yolks from the whites; reserve both separately. With an electric beater, or by hand, combine the yolks with 1/2 cup of the granulated sugar and beat until thick and pale. Mix in the cacao powder, vanilla, and salt to combine

Step 3

In the large, clean bowl of a stand mixer fitted with the whisk attachment, or by hand, whip the egg whites on medium-high until they form soft peaks. Gradually add the remaining 1/4 cup granulated sugar, and whip until stiff peaks form.

Step 4

Gently fold the egg whites into the yolk mixture, then spread the batter evenly onto the prepared baking sheet. Bake until cake begins to pull away from the sides of the pan and the top is dry, spongy, and springs back when gently pressed, 12 to 15 minutes.

Step 5

Dust a clean (non-terrycloth) dishtowel with the powdered sugar. Run a knife around the edges of the warm cake to loosen it from the pan, then turn the cake out onto the towel. Gently peel off and discard the parchment paper. (This can be tricky, as the cake is sticky, so go slow and steady and help the cake along with your fingers, if needed; don’t worry if a thin layer of cake sticks to the paper here and there. The frosting will cover any imperfections.) Roll up the cake, along with the towel, from the short end. Set the towel and cake roll on a wire rack for 30 minutes to cool.

Step 6

Make the frosting: Carefully open the coconut-milk cans. Scoop out only the firm, waxy solids at the top of each can and place them in the bowl of a stand mixer fitted with the whisk attachment (discard the remaining liquid or reserve for another use). Beat on medium speed for 30 seconds. Scrape down the sides of the bowl and beat on medium-high until the cream forms stiff peaks, about 3 minutes. Add the powdered sugar, cacao powder, vanilla, and salt. Mix slowly to incorporate, then beat on high speed until fluffy and completely combined. Keep refrigerated until use.

Step 7

Unroll the cake. Using a spatula, spread 2/3 of the chilled frosting in a thin layer on the cake, leaving a 1-inch border on all sides. Roll the cake into a log again, this time without the dish towel, so the frosting is inside the log.

Step 8

Gently transfer the cake to a serving platter seam side down, then use the remaining frosting to frost the outside of the log. Gently run a fork through the frosting to create a log-like texture. Cut off the ends of the log for a clean look.

Step 9

Rinse the rosemary and cranberries to moisten. Sprinkle with sugar. Let dry. Then use to decorate the cake. Keep cake refrigerated until serving to keep the frosting firm.