Mocha Yule Log

To prepare cake: Preheat oven to 350 degrees F. Line the bottom of a large (12-by-16 1/2-inch) rimmed baking sheet (half sheet pan) with parchment paper; coat the paper and pan sides with cooking spray. Place eggs (in the shell) in a stand mixer bowl or large mixing bowl, add very warm tap water and set aside to warm the eggs and bowl.

Melt butter in a small saucepan over medium-low heat, swirling occasionally, until the white flecks of milk solids in the bottom of the pan start to turn golden brown, 4 to 8 minutes. Scrape into a medium bowl. Let cool to room temperature, then add 2 teaspoons vanilla. Set aside.

Meanwhile, whisk whole-wheat flour, cake flour and 1/4 cup cocoa in a medium bowl; set aside.

Drain the water and break the eggs into the warmed mixing bowl. Add sugar and 1/2 teaspoon salt and beat with an electric mixer on medium-high speed until thick and pale light yellow, 5 to 15 minutes (depending on the power of your mixer). To test if it's beaten well enough, lift the beater from the batter: as the batter falls off the beater into the bowl, it should mound for a moment on the surface.

Gently fold the flour mixture into the egg mixture with a whisk, in two additions, until just incorporated. Gently fold about 1 cup of the batter into the reserved butter. Then gently fold the butter mixture into the bowl of batter with a whisk until just incorporated, being careful not to overmix. Spread the batter evenly in the prepared baking sheet, spreading completely to the sides.

Bake the cake until puffed and a toothpick inserted in the center comes out with a few moist crumbs attached, 8 to 12 minutes. Cool in the pan on a large wire rack for 10 minutes. Gently run a knife around the edges and turn the cake out onto the rack; remove the parchment and let cool completely. Once cool, cover with 2 overlapping pieces of plastic wrap and a clean, damp kitchen towel to prevent it from drying out. (The cake can be held this way for up to 4 hours before assembling the Yule Log.)

To prepare filling & frosting: Bring 2 inches of water to a simmer in the bottom of a double boiler. (No double boiler? See Tips.) Combine agave nectar, instant coffee, reconstituted egg whites, cream of tartar and 1/4 teaspoon salt in the top of the double boiler. Beat with an electric mixer on medium speed until well combined, about 1 minute. Place over the simmering water and beat on high speed until the frosting is glossy and has the texture of very thick shaving cream, 5 to 10 minutes. Remove from the heat and beat in 1 teaspoon vanilla until just combined.

Leaving the towel and plastic wrap over the cake, invert it onto a work surface with a long edge nearest you. The towel will now be on the bottom, with the plastic wrap directly beneath the cake. Combine coffee and half-and-half in a small bowl. Brush the top of the cake with the coffee mixture; let it soak in and continue brushing on more until all of it is absorbed.

Spread about two-thirds of the frosting evenly over the cake. Using the plastic wrap, lift the long edge and roll the cake into a log lengthwise. Cut a 3 to 4-inch "branch" off one end at an angle. Place the longer log on a serving platter, seam-side down. Use a little frosting to attach the branch to the main log. Cover the cake and branch with the remaining frosting. Make decorative ridges in the frosting with a fork to resemble bark. Let the cake stand at room temperature for at least 30 minutes or up to 2 hours. Or refrigerate, uncovered, for up to 1 day.

Just before serving, garnish the serving platter with Sugared Rosemary and Meringue Mushrooms, if desired.