To make cake: Set oven rack in middle position; preheat oven to 350 °F. Coat a 15-by-10-inch jelly roll pan with cooking spray. Dust with flour, shaking out excess. Line bottom with parchment or wax paper. Spray paper with cooking spray.
Combine flour, hazelnuts, baking powder and 1/4 teaspoon salt in a food processor or blender; process until the hazelnuts are ground.
Beat whole eggs and 1/4 cup granulated sugar with an electric mixer until thickened and pale, about 5 minutes. Blend in 1 1/2 teaspoons vanilla.
Beat 3 egg whites in a clean mixing bowl with an electric mixer until soft peaks form. Gradually beat in the remaining 1/4 cup granulated sugar, beating until stiff and glossy. (If you have a stand-up mixer and a portable mixer, you can beat the whole egg mixture and the egg whites simultaneously.) Whisk one-fourth of the beaten whites into the whole-egg mixture. Sprinkle half of the flour mixture over top and, with a rubber spatula, fold just until blended. Fold in the remaining beaten whites, then the remaining flour mixture.
Turn the batter into the prepared pan, spreading to the edges. Bake until the top springs back when touched lightly, 12 to 15 minutes. (Do not overbake or the cake will crack when rolled.) While the cake is baking, place a clean kitchen towel on the work surface. Cover it with parchment or wax paper, coat the paper with nonstick cooking spray and dust liberally with cocoa.
Once the cake is done, loosen the edges and invert the cake onto the cocoa-dusted paper. Peel the paper from the top of the cake. Trim the edges of the cake with a serrated knife, then quickly roll up the cake in the paper-lined towel, starting at a short end. Set the rolled cake, seam-side down, on a rack to cool.
To make filling and frosting: Dissolve instant coffee in boiling water and set aside. Bring about 1 inch of water to a simmer in the bottom of a large saucepan. Combine 2 egg whites, 3/4 cup granulated sugar, cream of tartar, pinch of salt and water in a heatproof mixing bowl that will fit over the saucepan. Set the bowl over the simmering water and beat with an electric mixer at low speed until an instant-read or candy thermometer registers 140 °F, 3 to 5 minutes. Increase speed to high and continue beating over the heat for 3 minutes. The mixture should be shiny and form soft peaks. Remove the bowl from the heat and beat until cool and fluffy, about 4 minutes. Set aside.
Spoon 1 cup of the frosting into a small bowl. Add cocoa and 1 teaspoon vanilla; whisk until smooth. Add the dissolved coffee and coffee liqueur to the remaining frosting and beat with an electric mixer until smooth and thick.
To assemble and decorate buche: Unroll the cake and brush the surface with hazelnut liqueur. Spread with the chocolate filling. Roll up again as tightly as possible (without the paper or towel) and place, seam-side down, on a serving platter. Slip strips of wax paper under cake edges. Spread the coffee frosting over cake. (There will be some frosting left over.) Make decorative ridges in the frosting with a fork and remove strips of wax paper.
Roll rosemary sprigs and cranberries in confectioners' sugar to create a frosted effect. If the sugar will not cling, lightly mist the rosemary and cranberries with water. Arrange the frosted rosemary sprigs and cranberries decoratively around the cake. Nestle meringue mushrooms among the sprigs. Top the cake with chocolate curls and dust with confectioners' sugar.
Make Ahead Tip: The cake (Steps 1-7) can be made up to 1 month ahead and frozen, towel and all, wrapped in a plastic bag. Thaw at room temperature for at least 1 1/2 hours before unrolling. The frosted cake can be held, loosely covered, in the refrigerator overnight.
Tips: To toast hazelnuts: Spread nuts on a baking sheet and bake at 350°F, stirring once, until fragrant, 7 to 9 minutes.
To bring an egg to room temperature: Either set it out on the counter for 15 minutes or submerge it (in the shell) in a bowl of lukewarm (not hot) water for 5 minutes.