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1.
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Heat oven to 425.
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To make the puffs: In a medium saucepan, melt butter in 1 1/2 cups water with salt and sugar.
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Remove pan from heat, and add flour.
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Return pan to heat and, using a wooden spoon, beat vigorously for 2 to 3 minutes.
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(A film should form on the bottom of the pan.)
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Cool slightly, and add 6 eggs, one at a time, beating vigorously.
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2.
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Make a glaze by beating the remaining egg with 1 teaspoon water, and set aside.
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Using a pastry bag fitted with a coupler and a 1/2-inch-wide plain tip, pipe out mounds that are 1 inch high and 3/4 inch in diameter on a parchment-lined baking sheet.
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Brush with egg glaze, and smooth the tops.
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Bake until puffed and golden, 20 to 25 minutes.
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Cool on racks.
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(The puffs can be made ahead and frozen until ready to assemble.)
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3.
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Make the pastry cream: In the bowl of an electric mixer fitted with the whisk attachment, beat egg yolks, gradually adding sugar, until mixture is thick and pale yellow.
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Beat in flour.
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Scald milk, and add in dribbles to egg mixture, reserving 1/2 cup.
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Place mixture in a clean pot over high heat, and stir vigorously until mixture boils and thickens.
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If it seems too thick to pipe, add reserved milk.
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Remove from heat.
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Using a hand whisk, beat butter into egg mixture, one tablespoon at a time.
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4.
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In a double boiler or heat-proof bowl set over simmering water, melt chocolate and espresso together until smooth.
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Add chocolate mixture to the pastry cream; let cool completely.
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Just before assembling croquembouche, fill a pastry tube fitted with a 1/4-inch-wide tip with pastry cream, insert tip into puffs, and pipe in cream to fill.
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5.
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To make the caramel: In a medium saucepan, combine 2/3 cup water, sugar, and corn syrup, and bring to a boil over high heat.
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Do not stir.
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Cover pan, and boil until steam dissolves any crystals.
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Uncover, and boil 5 more minutes, or until syrup is amber in color.
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Remove from heat.
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Dip the bottom of each puff into the caramel, and arrange puffs in a pyramid.
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6.
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To make a spun-sugar web to wrap around the croquembouche: Cut the looped ends of a wire whisk with wire cutters, or use two forks held side by side, and dip the ends into caramel.
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Wave the caramel back and forth over the croquembouche, allowing the strands to fall in long, thin threads around it.
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Wrap any stray strands up and around the croquembouche.
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Serve.
