Cardamom-Pistachio Bûche de Noël

Cardamom Meringue Mushrooms:
2 large egg whites, room temperature (reserve yolks)
1/4 tsp. cream of tartar
1/4 tsp. kosher salt
1/2 cup granulated sugar, divided
1/4 tsp. ground cardamom
Unsweetened cocoa powder (for dusting)
2 oz. chopped bittersweet chocolate
Pistachio Filling:
6 oz. raw pistachios
1 3/4 cups whole milk
1 1/2 tsp. unflavored powdered gelatin
5 large egg yolks (from meringues and frosting)
1/2 cup granulated sugar
3/4 tsp. kosher salt
1/4 tsp. almond extract
Chocolate Buttercream Frosting:
3 large egg whites (reserve yolks)
2/3 cup granulated sugar
1/4 tsp. kosher salt
8 oz. chopped bittersweet chocolate
1 cup (2 sticks) unsalted butter, room temperature, cut into pieces
1 1/2 tsp. vanilla extract
Cardamom Sponge Cake and Brandy Syrup:
Neutral oil or unsalted butter (for pan)
6 large eggs, room temperature, separated
1 3/4 cups granulated sugar, divided
1 tsp. vanilla extract
1 1/2 cups cake flour
1/2 tsp. baking powder
1/2 tsp. kosher salt
1/2 tsp. ground cardamom, divided
1 tsp. cream of tartar
Powdered sugar (for dusting)
1/3 cup brandy
Chocolate Bark and Assembly:
8 oz. chopped dark chocolate
2 Tbsp. unsweetened cocoa powder
2 oz. raw pistachios, finely ground
Powdered sugar (for dusting)

Cardamom Meringue Mushrooms:
Preheat oven to 250°F. Using an electric mixer on high speed, beat egg whites, cream of tartar, and salt in a large bowl until foamy and the yellowish tinge has disappeared, about 2 minutes. With the motor running, add granulated sugar about 1 Tbsp. at a time until mixture is very stiff and shiny, about 5 minutes.
Transfer meringue to a piping bag fitted with a 1/2" round pastry tip (or a freezer bag with a 1/2" opening cut from one bottom corner). Pipe 12 mounds ranging in size from 1"–1 1/2" in diameter for the mushroom caps on a parchment-lined rimmed baking sheet. Pipe 12 cones straight up about 1 1/2" high for the mushroom stems. With a lightly damp finger, tamp down points on rounds. Sprinkle rounds with cocoa powder. Pipe extra tops and stems with remaining meringue as backups.
Bake meringues until dry and set, about 2 hours. Open oven door 10 minutes to release any steam. Turn oven off, shut door, and let meringues dry out completely, about 2 hours longer or up to overnight.
When ready to assemble, melt chocolate in a medium heatproof bowl in microwave, stirring every 10 seconds, until melted, about 20 seconds. Using an offset spatula, release meringues from parchment. Using a paring knife, make a small hole in the center of flat side of each mushroom cap. Spread some chocolate on flat side, then stick top of stem into hole. Return to parchment and repeat with remaining tops and stems. Let mushrooms stand at room temperature until chocolate is firm.
Pistachio Filling:
Combine pistachios and milk in a small saucepan. Bring to a simmer over medium heat and cook, stirring occasionally, until pistachios are very tender, about 20 minutes. Remove and skim off any loose pistachio skins.
Meanwhile, sprinkle gelatin over 2 Tbsp. water in a small bowl and let stand 5 minutes.
Transfer pistachio mixture to a food processor and blend until smooth. Return mixture to saucepan and whisk in egg yolks, granulated sugar, and salt. Cook over medium-low heat, stirring, until mixture begins to thicken but do not let boil, 3–5 minutes. Remove from heat and stir in gelatin mixture and almond extract. Transfer to a small bowl and let cool.
Chocolate Buttercream Frosting:
Whisk egg whites, granulated sugar, and salt in a large heatproof bowl. Set bowl over a pot of simmering water (do not let bowl touch water) and cook, whisking constantly, until mixture becomes fluffy and opaque and you can’t feel any grains of sugar when you rub mixture between two fingers, about 5 minutes. Remove bowl from heat.
Melt chocolate in a medium heatproof bowl in microwave, stirring every 20 seconds, until melted, about 1 minute. Let cool slightly.
Using an electric mixer on high speed, beat egg white mixture until cool, glossy, and fluffy, about 10 minutes. (A stand mixer fitted with the whisk attachment makes this job much easier!) Add butter one piece at a time, beating until fully incorporated before adding the next, until fluffy and smooth, another 4 minutes. If mixture looks curdled, switch to the paddle attachment and/or apply a little bit of heat to underside of bowl with a washcloth dipped in hot water and continue to beat until smooth. Beat in chocolate and vanilla extract until combined.
Do Ahead: Frosting can be made 3 days ahead. Transfer to an airtight container and chill. Bring to room temperature and beat to desired consistency.
Cardamom Sponge Cake and Brandy Syrup:
Place a rack in center of oven; preheat to 375°F. Grease an 18x13" rimmed baking sheet with oil. Line with parchment paper and grease parchment.
Using electric mixer on high speed, beat egg yolks and 1/4 cup water in a large bowl until doubled in size and thick and fluffy, about 5 minutes. With the motor running, gradually add 1 cup granulated sugar, a little at a time, until mixture is very thick. Beat in vanilla.
Sift flour, baking powder, salt, and 1/4 tsp. cardamom over egg yolk mixture and fold gently to combine (it will be very thick).
Using electric mixer on high speed, beat egg whites and cream of tartar in another clean large bowl until foamy and the yellowish tinge has disappeared, about 1 minute. Beat in 1/2 cup granulated sugar, about 1 Tbsp. at a time, until shiny, medium stiff peaks form. Fold about one-quarter of egg white mixture into egg yolk mixture to loosen. Fold remaining egg white mixture into egg yolk mixture until combined.
Transfer batter to prepared pan; smooth top. Bake cake until puffed and set and springs back when gently poked in the center, 15–20 minutes. Transfer cake to a wire rack and let cool 5 minutes.
Set a clean dish towel on a work surface and dust completely with powdered sugar. Cut around edges of cake with a sharp paring knife. Invert cake onto prepared towel; remove parchment. Without squishing the cake and starting on long side, gently roll up cake in towel. Transfer cake to a wire rack and let cool completely.
While cake cools, make the brandy syrup. Cook remaining 1/4 cup granulated sugar and 1/4 tsp. cardamom with 2 Tbsp. water in a small saucepan over medium heat, stirring occasionally, until sugar is dissolved, about 2 minutes. Remove from heat and stir in brandy. Let cool.
Chocolate Bark and Assemble:
Melt chocolate in a medium heatproof bowl in microwave, stirring every 20 seconds, until melted, about 1 minute. Using an offset spatula, spread melted chocolate across a parchment-lined rimmed baking sheet. Sift cocoa over and freeze bark until solid, about 10 minutes.
Gently unroll cake. With a pastry brush, evenly brush with brandy syrup until all syrup is absorbed. Spread filling evenly over cake and roll it back up without towel.
Transfer to a platter. Trim ends. Cut off about 3" from both ends on a diagonal. Place angled side of each small piece against roll to create branches, positioning one on top and the other on the side, using a large dab of frosting to adhere. Cover any exposed cake on sides with more frosting.
Break chocolate bark into long shards and press into cake in a decorative pattern. Top and surround cake with meringue mushrooms. Use pistachios to create “moss” on and around log. Dust with powdered sugar.