Position a rack in the lower third of the oven and preheat to 375 degrees . Line a 9-inch springform pan with parchment paper. Wrap heavy-duty foil around the bottom and sides of the pan and press tightly to seal.
In a medium saucepan, combine the espresso, brown sugar and cinnamon sticks and simmer over low heat until fragrant, about 20 minutes. Discard the cinnamon sticks. Add the butter and cayenne and stir until the butter is melted. Pour into a large bowl and set aside.
Coarsely chop the semisweet and unsweetened chocolate and whisk into the coffee mixture until melted; whisk in the eggs. Scrape the batter into the prepared pan. Place the pan inside a larger ovenproof pan and pour in enough water to reach halfway up the sides. Bake until a toothpick inserted in the center comes out dry, 50 minutes to 1 hour.
Carefully remove the pan from the water bath, set on a rack to cool slightly, then refrigerate for at least 8 hours. Run a wet knife around the edge of the cake, then remove the foil and release the springform. Place a platter over the cake and invert. Remove the bottom of the pan and discard the parchment. Slice the cake and serve with the berries.
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