Make the meringue decorations: Heat oven to 200°. Place 1⁄3 cup sugar and 2 egg whites in a bowl set over a saucepan of simmering water; stir mixture until egg whites register 140° on an instant-read thermometer. Remove bowl from saucepan and beat with a hand mixer on high speed until cooled. Place 1 cup meringue in a bowl, and stir in food coloring; transfer green meringue to a piping bag fitted with a 3⁄8-inch star tip. Working on a parchment paper-lined baking sheet, pipe two 1 1⁄2-inch-wide stars; pipe a 1-inch-wide star on top of each larger star, and then pipe a 1⁄2-inch-wide star on top of each middle star to form a three-tiered Christmas tree. Transfer uncolored meringue to a piping bag fitted with a 3⁄8-inch-round tip; pipe four 1 1⁄2-inch-wide mounds to resemble mushroom caps, and then pipe four 1⁄2-inch-wide x 1 1⁄2-inch-tall sticks to resemble mushroom stems. Bake meringue shapes until dry and crisp, about 2 hours. Turn off oven and let shapes cool completely in oven.
Make the ganache icing: Place chocolate in a bowl; set aside. Bring cream and honey to a boil in a 2-qt. saucepan over medium-high heat; pour over chocolate and let sit for 1 minute. Using a rubber spatula, slowly stir cream and chocolate until smooth and shiny; let cool at room temperature until set and thick, at least 6 hours or overnight.
Make the coffee buttercream filling: Place 1 cup sugar and 4 egg whites in the bowl of a stand mixer and set it over a saucepan of simmering water; stir mixture until egg whites register 140° on an instant-read thermometer. Remove bowl from saucepan and place on stand mixer fitted with a whisk; beat on high speed until meringue is cooled and forms stiff peaks. Replace whisk with paddle and add butter to meringue; beat until smooth, stir in espresso, and set aside.
Make the sponge cake: Heat oven to 400°. Grease and flour a 13-inch x 18-inch rimmed baking sheet, lined with parchment paper, and set aside. In the bowl of a stand mixer fitted with a whisk, beat 2⁄3 cup sugar and eggs on high speed until mixture falls back in thick ribbons when lifted from the whisk, about 6 minutes; fold in butter and flour. Spread batter into an even layer on bottom of prepared baking sheet and bake until golden brown on the bottom, about 12 minutes. Place a clean kitchen towel that is larger than the baking sheet on a work surface, and dust it liberally with confectioners' sugar. Invert cake onto towel; dust with more sugar. Starting with a narrow end of the rectangle, immediately roll cake up into a jelly roll, letting the towel roll inside the cake. Let cool to room temperature.
Make the rum syrup: Bring 2 tbsp. sugar, rum, and 1 tbsp. water to a boil in a 1-quart saucepan over high heat; cook until sugar dissolves and set aside to cool.
Assemble the cake: Once cooled, gently unroll and remove towel from cake. Brush the inside with the rum syrup; allow to soak in for about 2 minutes. Spread buttercream evenly over cake; re-roll cake and set the roll seam side down on a serving platter. Slice about 3 inches off one end of the cake roll at a 30° angle; cut the other end to make it flat. Spread the flat end of the angled slice with a little buttercream and set the slice on top of the cake roll to create a "stump." Stir ganache until smooth and, using a small offset spatula, spread ganache over cake, leaving the ends of cake and cut top of the "stump" exposed. Drag the tines of a fork along the ganache, making markings to resemble bark; refrigerate until chilled.
Decorate the Bûche de Noël: Using ganache as glue, place meringue "caps" on top of "stems" to form mushrooms. Dust cocoa powder lightly over the mushrooms, and sprinkle gold dust lightly over the entire Bûche de Noël. Place meringue mushrooms and Christmas trees decoratively on and around the Bûche de Noël before serving.