1½ teaspoons strong espresso (either from a shot or from powdered espresso)
2 sticks unsalted butter, room temperature
1 cup sugar
For the cake
Preheat the oven to 325 degrees. Line the bottom of a 12×17-inch sheet pan or jelly roll pan with parchment paper cut to fit exactly. Do not grease the sides of the pan.
Sift together the flour and baking powder into a large bowl. Add all sugar except the 3 tablespoons of sugar you have set aside. Add salt and whisk to combine.
In a small bowl whisk together oil, egg yolks, water, and vanilla.
Make a well in the flour and sugar mixture, add the egg yolk mixture, and then whisk thoroughly and quickly for about a minute until very smooth.
Place the egg whites in a large mixing bowl and using a mixer fitted with the whisk attachment beat on medium speed until frothy. Add the cream of tartar (or lemon juice) and beat on medium-high speed until the whites hold soft peaks. Slowly add the 3 tablespoons sugar and beat on medium-high until the whites hold firm, shiny peaks.
Using a rubber spatula, scoop out about one-third of the whites onto the yolk mixture and fold in gently to lighten the batter. Gently but thoroughly fold in the remaining whites just until combined.
Pour the batter into the prepared pan, smoothing the top with an offset spatula. Bake until the cake is just set to the touch 20−30 minutes. Let cool and set aside. You can make the cake ahead and store in the refrigerator well wrapped for up to 3 days.
For the Chocolate Frosting
Put the chocolate in a heat-proof bowl set over a simmering saucepan of water (do not let the bottom of the bowl touch the water) and stir occasionally until the chocolate is completely melted. Set aside and let cool to room temperature.
In the bowl of a stand mixer fitted with the paddle attachment (or in a medium mixing bowl using a hand-held mixer) beat the butter on medium speed until smooth and creamy.
Add the milk until completely blended. Add the cooled chocolate and mix until completely incorporated, 2−3 minutes, scraping down the sides of the bowl with a rubber spatula as necessary. Add the vanilla and beat just until mixed.
With the mixer on low speed, gradually add the confectioners sugar until you reach a creamy and silky frosting consistency. The frosting can be stored in an airtight container at room temperature for up to 2 days.
For the Coffee syrup
With the coffee still hot, add sugar until dissolved.
For the Sugared rosemary and cranberries
Make a simple syrup using ¾ cup sugar and ½ cup water. In a saucepan bring to a boil and set aside to cool.
Once the syrup is cool, drop in rinsed fresh rosemary and cranberries and drain with a slotted spoon.
Roll in extra fine sugar and set aside to dry.
For the Pistachio moss
Put the pistachios in a small blender or coffee grinder and pulse until they look like powder “moss.”
For the Espresso buttercream frosting
Combine the flour and ¼ cup of the milk, vanilla, and 1½ teaspoons espresso in a small heavy-bottomed saucepan and whisk until blended. Set the pan over medium heat and gradually add remaining ¾ cup milk, whisking constantly. Cook the mixture, whisking, until it comes to low boil. Then reduce the heat to low and continue to whisk until the mixture begins to thicken and starts to “burp,” 2−3 minutes. Remove from heat.
Transfer the mixture to a small heat-proof bowl and stir it occasionally as it cools to keep it free of lumps. (If you do get a few lumps, don’t worry—you can whisk the mixture to dissolve the lumps or pass it through a fine-mesh sieve.) Set the mixture aside to cool to room temperature. (You can put it in the refrigerator for 10 minutes to speed up the process.)
In standing mixer, add softened butter and beat until fluffy, about 3 minutes. Then add sugar and beat for 3 more minutes until mixture is light and thick. Add cooled milk/flour mixture, and beat for 4−5 minutes until it is thickened and creamy.
Whip at high speed for 1 to 2 minutes. Cover with plastic wrap and keep in refrigerator up to 3 days.
To assemble the yule log
Place the cake on a piece of parchment or plastic wrap. Using a pastry brush, moisten the cake with the coffee syrup.
Spread buttercream evenly over the cake. Leave a 1-inch strip of cake without buttercream on the long sides.
Starting from the long side nearest you, begin rolling up the cake, using the parchment of plastic wrap to help you lift and roll evenly and tightly. Don’t worry about how the ends look—they will be cut off. Refrigerate the rolled cake until the buttercream is firm, about 2 hours.
Using a sharp knife, cut a thick slice on the diagonal off each end of the log and set aside. To frost the log, place it seam side down. Use a small offset spatula to place dollops of the frosting along the top. Choose the prettiest cut-off end and place it slanted side up on the log to create a “bough.” Frost the sides of the “bough.”
Transfer the cake to a serving plate. Be creative and finish decorating with moss, sugared cranberries and rosemary, and other decorations.