Bûche de Noël

Ingredients:
6 large egg whites
1 1/4 cups sugar
1/4 teaspoon cream of tartar
1/4 teaspoon kosher salt
Unsweetened cocoa powder (for dusting)
1 cup (2 sticks) unsalted butter, cut into 1" pieces, room temperature
1 teaspoon vanilla extract
1 vanilla bean, split lengthwise
Nonstick vegetable oil spray
1/4 cup all-purpose flour
3 tablespoons cornstarch
1/3 cup unsweetened cocoa powder, plus more for dusting
1/4 cup whole milk
3 tablespoons unsalted butter, cut into pieces
1 tablespoon vegetable oil
1 teaspoon vanilla extract
1/2 teaspoon kosher salt
3 large eggs, room temperature
3 large egg yolks, room temperature
3/4 cup sugar
Cocoa Syrup
Bittersweet Ganache
Mascarpone Filling
1 ounce marzipan (about 2 tablespoons)
2 tablespoons dark unsweetened cocoa powder, preferably Dutch-process
1 ounce bittersweet chocolate
melted

Instructions:
Preheat oven to 225°F. Whisk egg whites, sugar, cream of tartar, and salt in a large heatproof bowl set over a medium saucepan of simmering water (bowl should not touch water) until egg whites are very liquid (mixture will be warm to the touch) and sugar is dissolved (rub between your fingers to check), about 5 minutes. Remove bowl from heat and beat with an electric mixer until stiff, glossy peaks form (bowl will feel cool), about 5 minutes.
Line a rimmed baking sheet with parchment paper and dust very lightly with cocoa powder. Scoop out 1 cup meringue. Dollop heaping teaspoonfuls of meringue onto prepared baking sheet, spacing at least 1 1/2" apart.
Lightly dust meringue with more cocoa powder and place a sheet of parchment over top. Use an offset spatula or your hands to flatten meringue into irregularly shaped disks about 1/8" thick. Bake until lightly browned and parchment peels away easily, 75–90 minutes (meringue will be soft when it first comes out of the oven but will dry and crisp as it cools).
Meanwhile, with mixer on medium speed, beat butter into remaining meringue, adding a piece at a time and waiting until incorporated before adding more. Beat until mixture is very smooth. (If it looks very loose or curdled, just keep beating; buttercream will come back together.) Add vanilla extract and scrape in vanilla seeds; beat to combine.
Mushrooms and buttercream can be made 2 days ahead. Store mushrooms airtight at room temperature. Cover and chill buttercream; bring to room temperature, then beat until smooth before using.
Increase oven to 400°F. Coat a 26x18" rimmed baking sheet with nonstick spray and line with parchment paper, leaving overhang on long sides. Spray parchment. Whisk flour, cornstarch, and ⅓ cup cocoa powder in a small bowl.
Bring milk, butter, oil, vanilla, and salt to a simmer in a small saucepan over medium-low heat. Keep warm over low heat.
Meanwhile, beat eggs and egg yolks with an electric mixer on medium speed until smooth. Increase speed to high; beat until doubled in volume. With motor running, gradually add sugar; beat until very light and fluffy and mixture falls back on itself in a slowly dissolving ribbon (it should be at least quadrupled in volume), about 5 minutes.
Reduce speed to medium and gradually stream in milk mixture. Sift one-third of dry ingredients over top; gently fold in until only a few streaks remain. Working in 2 additions, repeat with remaining dry ingredients, scraping bottom of bowl and using as few strokes as possible to keep eggs from deflating (a few streaks are fine). Scrape batter into prepared baking sheet and gently spread to edges of pan. Tap sheet lightly on counter to pop any large air bubbles.
Bake cake until surface is puffed and springy to the touch, 10–12 minutes.
Let cake cool in pan 2 minutes, then run a knife along short edges to loosen. Invert onto a wire rack and carefully peel away parchment. Using a fine-mesh sieve, dust cake with cocoa powder. Cover with a large kitchen towel. Place another wire rack on top and flip cake over so towel side is underneath. Remove top rack; dust exposed side with cocoa powder. Starting at one of the long sides, gently roll up warm cake inside towel. Let cake cool, seam side down, 30–35 minutes.
Cake can be baked 1 day ahead. Store tightly wrapped in plastic at room temperature.
Carefully unroll towel and cake on a flat surface (cake will curl at the ends and may have a few small cracks but should stay in 1 piece). Position cake so the end that was in innermost part of spiral is closest to you; brush off any excess cocoa.
Using a pastry brush, gently dab cocoa syrup over entire surface of cake; you may not use it all. Dollop bittersweet ganache over top and smooth with an offset spatula to create an even layer.
Dollop mascarpone filling over ganache; carefully spread over surface, trying not to blend with ganache and avoiding last inch of cake along long side farthest from you.
Using towel to lift edge nearest you, reroll cake, keeping towel on exterior. Chill, seam side down, until filling is set, about 30 minutes.
Transfer filled cake to a baking sheet. Set aside 1/2 cup buttercream for attaching branches. Evenly spread remaining buttercream over cake with an offset spatula.
Using a long serrated knife, trim 1/2" of cake from each end to create clean edges; discard (or eat!). Slice off a 4" piece of cake. Starting 1" from end, divide 4" piece in half, cutting at a 45° angle, leaving 1" at opposite end. Transfer log to a platter.
Place angled side of each small piece of cake against roll to create branches, positioning one on top and the other on the side using a large dab of buttercream to secure. Cover any exposed cake on sides with more buttercream but leave cut ends exposed.
Use spatula to create textured lines in buttercream to look like birch bark.
Knead together marzipan and cocoa powder on a surface until smooth. Roll out on a sheet of parchment paper to less than 1/8" thick, then cut out wavy strips to look like tree knots; drape over buttercream and press gently to adhere.
Using either a pastry bag fitted with a very small round tip or a disposable plastic bag with one corner snipped off, drizzle melted chocolate over log to mimic birch bark striations. Chill Bûche de Noël, uncovered, until ready to serve.
Just before serving, press mushrooms perpendicularly into log in groups of 2 or 3.
Bûche de Noël (without meringue mushrooms) can be assembled 1 day ahead. Wrap in plastic and chill. Attach mushrooms just before serving.