Make the cake
Position a rack in the bottom third of the oven and heat the oven to 350°F.
Butter an 18x13-inch rimmed baking sheet. Line the pan with parchment. Butter the parchment and sprinkle 2 Tbs. of the cocoa over it, shaking the pan for even coverage and knocking out the excess.
In a double boiler, melt the chocolate with the coffee, whisking until smooth. Remove from the heat.
In a stand mixer fitted with the whisk attachment, beat the egg yolks on medium speed, scraping the bowl as needed, until light in color and beginning to thicken, 2 to 3 minutes. Add 1/2 cup of the sugar and beat until very thick and pale yellow, about 2 minutes. Reduce the speed to low and mix in the melted chocolate. Stop the mixer and sift 6 Tbs. of the cocoa over the mixture. Add the salt and stir with a rubber spatula until blended. Transfer to a large bowl.
In a clean stand mixer bowl with a clean whisk attachment, beat the egg whites on medium speed until frothy, about 30 seconds. In a steady stream, add the remaining 1/2 cup sugar. Increase the speed to medium high and whip until medium peaks form, 2 to 3 minutes.
With a rubber spatula, fold half the whites into the chocolate mixture to lighten it. Gently fold in the remaining whites until the batter is evenly colored, with no streaks of white. Transfer the batter to the prepared baking sheet, gently spreading it with an offset spatula.
Bake, rotating the pan halfway through, until the cake springs back when touched and is beginning to pull away from the sides of the pan, 18 to 22 minutes. Immediately loosen the cake from the sides of the pan with a paring knife. Sift the remaining 2 Tbs. cocoa evenly over the cake.
While the cake is still hot, lay a clean, unscented, lint-free kitchen towel over it. If the towel is long, extend one end about 1-1/2 inches beyond a short side of the cake and don’t worry about the other end. Invert a large rack or cutting board over the towel. Holding both the baking sheet and the rack with protected hands, invert the cake. Remove the baking sheet and parchment.
Using both hands and starting from the short end with the shortest towel overhang, roll the cake and the towel up together. Let cool for 30 minutes.
Carefully unroll the cake; it should look wavy and both ends should curl. (It’s OK if there are some small cracks.) Let cool completely, 10 to 15 minutes.
Using an offset spatula, spread the filling over the cake to within 1-1/2 inches of the far short edge and to within 1/2 inch of the other edges. Be sure to coax the filling into the interior of the closer curled end so that when the cake is sliced there will be filling in the center of the spiral.
Reroll the cake without the towel this time. The filling may squish out of the ends a bit, which is fine.
Finish the cake
Line a rimmed baking sheet with foil. If the cake is already on a rack from rolling, set that rack over the baking sheet. If it’s not, set a rack over the baking sheet and, using 2 large spatulas, transfer the cake to the rack.
Combine the cream and the corn syrup in a small saucepan and bring to a gentle simmer over medium heat. Remove from the heat and add the chocolate and butter. Let sit for 1 minute, then whisk until smooth. Let cool until fairly warm to the touch, then pour the glaze slowly and evenly over the roulade. If necessary, use a small spatula to spread the glaze on the sides of the cake. (Don’t worry about covering the ends; they will be trimmed later.) Garnish the roulade with white chocolate shavings, if you like.
Refrigerate the roulade uncovered on the rack for at least 1 hour and up to 8 hours to allow the glaze to set.
Using a knife with a thin blade dipped in hot water and wiped dry, trim the ends of the cake. Loosen the roulade from the rack by sliding the knife along the roulade where it’s been “glued” to the rack by the glaze. Transfer the cake to a serving platter. To serve, slice the cake using the same knife dipped in hot water and wiped dry after each slice.
The cream and white chocolate can be combined for the filling up to 24 hours ahead and refrigerated, covered. Whisk to medium peaks just before filling the cake.
The cake can be made, filled, and glazed up to 8 hours ahead. Refrigerate uncovered on the rack, and trim the ends just before serving.