Step 1

Preheat the oven to 350°F. Arrange a rack in the middle of the oven and heat to 350°F.

Step 2

Prepare the springform pan. Brush 1 tablespoon of the butter on the bottom and sides of a 9-inch springform pan. Sprinkle with 1 tablespoon of the cocoa powder. Holding the pan over the sink, move it in a circular motion to distribute the cocoa evenly over the buttered surfaces, discarding any excess into the sink; set aside.

Step 3

Melt the chocolate. Place the remaining 10 tablespoons butter and chopped chocolate in a large microwave-safe bowl. Microwave on the high setting in 15-second bursts, stirring between each burst, until the butter and chocolate are completely melted, 45 to 60 seconds total. Set aside to cool until the chocolate is warm, but not hot, 2 to 3 minutes.

Step 4

Combine the chocolate and other ingredients. Whisk the egg yolks into the cooled chocolate mixture. Sift in the remaining 2/3 cup cocoa powder, espresso powder, and salt. Add the vanilla and whisk until completely combined.

Step 5

Begin to beat the egg whites. Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment. (Alternatively, use an electric hand mixer and large bowl.) Beat on low to medium speed until the egg whites are broken up and foamy, 15 to 20 seconds.

Step 6

Add the cream of tartar. Add the cream of tartar and beat on medium to medium-high speed until the whites form fluffy, droopy peaks when the beater is lifted, 45 to 90 seconds.

Step 7

Beat to form stiff peaks. With the mixer on medium speed, gradually add the sugar and beat until you've created glossy, stiff peaks that stand by themselves when you lift the beaters slightly, 1 to 2 minutes. The exact timing will depend on the size and shape of the bowl, the power of your mixer, and other factors.

Step 8

Fold the whites into the batter. Whisk about a 1/4 of the whipped egg whites into the chocolate mixture. Then, with a silicone spatula, gently scoop the remaining egg whites into the batter. With the same spatula, scoop them from underneath, along with some of the batter, and very gently turn the whole thing over; repeat until just incorporated on no streaks of egg whites remain.

Step 9

Bake the cake. Pour the batter into the prepared pan and smooth the top with a silicone spatula. Bake until the cake has risen ever-so-slightly in the center and is set enough in the center that it will not jiggle when the pan moves; a cake tester will not come out clean, 34 to 37 minutes. The top will have formed an extremely thin, but discernible, crust. The cake will pull away from the side of the pan while it cools; don't let it bake until it pulls away from the side of the pan while in the oven or it will be overdone. Cool completely on a wire rack for at least 1 1/2 hours. Remove the side of the pan and serve with melted chocolate and ice cream if desired.