Heat oven to 350 degrees F.
Stand cones on an ungreased baking sheet. In a mixing bowl, whisk together the flour, baking soda, and salt; set dry ingredients aside.
Set a medium saucepan full of water over medium heat; bring water to a boil. Reduce heat to medium high, and let water simmer. In the heatproof bowl of an electric mixer, combine brewed coffee, 2 1/2 ounces chocolate, and 1/2 cup (1 stick) butter. Set bowl over simmering water, stirring occasionally, until the butter and chocolate have melted. Remove bowl from heat; stir in 1 cup sugar.
Transfer bowl to mixer. Add the reserved dry ingredients, and beat until well combined. Add the egg and 1 teaspoon vanilla; beat to combine. Pour batter into the cones, filling each cone 3/4 full.
Transfer baking sheet to oven, and bake until domes form and a cake tester inserted into centers come out clean, 20 to 25 minutes. Remove from oven, and transfer to a wire rack to cool completely.
Combine the remaining 1/2 cup plus 2 tablespoons sugar, espresso, and cream in a heavy saucepan. Set over high heat, and bring to a boil, stirring constantly. Reduce heat to medium; simmer, without stirring, 5 minutes. Remove pan from heat; add remaining 2 1/2 ounces chocolate, stirring until melted. Stir in the remaining 1/4 cup (1/2 stick) butter and 1/2 teaspoon vanilla. Chill, stirring occasionally, until icing is cool enough to spread. Using an offset spatula, spread icing over ice-cream-cone tops. Serve.
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