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1
Melt the butter with the chocolate in the microwave or in a double boiler until just liquid. Set aside.
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2
Whisk the eggs and sugar in a heatproof bowl over boiling water until they are about room temperature. Let a droplet fall onto your wrist: if you don't feel it, that means you're at the right temperature.
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3
Remove from the heat. Beat until light, syrupy, and fluffy, about 10 minutes with a whisk or 2-3 minutes with a handheld mixer. Beat in the flour, cocoa powder, and salt. Stir to combine, then pour in the melted butter and chocolate. Fold in. Refrigerate for an hour or overnight.
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4
When ready to bake, heat the oven to 400u00b0F. Butter a 9-inch cake pan with removable sides. Cut a circle of parchment and lay it on the bottom; butter that too. Press the batter into the pan (you may need to use wet hands to press it in evenly). Bake for about 20 minutes until the top is firm and the sides are starting to crisp up. Cool in the pan for 5 minutes, then carefully remove the sides. Flip the cake, peel off the parchment, and transfer to a rack or a plate to cool.
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5
Whip the cream with the confectioners sugar and the 1/4 teaspoon vanilla until it holds soft peaks. Set aside in the fridge.
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6
Combine the butter, sugar, salt, and cream in a saucepan over medium heat. Bring to a boil slowly, stirring constantly, then simmer for 3-5 minutes, stirring constantly and making sure the mixture doesn't boil over. Remove from the heat and whisk in the vanilla and espresso. As soon as it's cool enough to taste, check for flavors, adding more salt, vanilla, or espresso as necessary.
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7
Cut thin slices of the Chocolate Madeleine Cake. Top with a few tablespoons of whipped cream and a spoonful of Coffee-Caramel Sauce, either warm or room temperature.
