Make the pate feuilletee: In the bowl of an electric mixer fitted with the paddle attachment, beat the butter until smooth. Add 1/2 cup flour; mix until smooth. Scrape this butter mixture onto a piece of waxed paper. Form into a 5-inch square, wrap; transfer to the refrigerator to chill. Place remaining flour in mixer bowl; add salt and 1 cup water. Mix carefully into a smooth dough, being careful not to over mix. Gather dough into a ball, and wrap in plastic. Chill a few minutes. On a lightly floured surface, roll out the dough into a 6 by-12-inch rectangle. Place the chilled butter square in the center of the rectangle. Fold the flour dough over the butter dough, completely encasing the butter. Press the edges of the dough together, sealing as well as possible. Transfer to a sheet of waxed paper; chill until dough and butter reach the same temperature, about 1 hour. On a lightly floured surface, roll out dough into a 3/8-inch-thick rectangle. Fold into thirds, like a letter, and roll into another 3/8-inch-thick rectangle. Transfer dough to waxed paper, and chill 1 hour. (Try to use as little flour as possible for the rolling, and brush off any excess before folding dough.) Repeat rolling-and-folding process 2 more times; chill 1 hour. Roll and fold 2 more times. When you fold the dough for the sixth time, the dough should be extremely smooth and silky, with no lumps of butter visible. (If the dough becomes too elastic, set aside in a cool place 15 to 20 minutes before rolling.) Divide dough in half. Wrap each half in plastic wrap. Refrigerate one half for the gateau Saint-Honore, and freeze the other half for another use.
Make the pate a choux: Line a baking sheet with parchment paper; set aside. Preheat the oven to 375 degrees F. Combine butter and 1 cup water in a small saucepan, and set over high heat. Bring to a boil, and immediately add flour and salt. Beat continuously with a wooden spoon until the dough comes away from the sides of the pan. Transfer the mixture to the bowl of an electric mixer fitted with the paddle attachment. Add eggs, 1 at a time, beating well after each addition. Set aside 1 1/2 cups dough for assembling the cake. Place remaining mixture in a pastry bag fitted with an Ateco #9824 tip. Pipe 1-inch balls, spaced 1 1/2 inches apart, on the prepared baking sheet. Bake balls until dark golden brown, 40 to 45 minutes. Transfer to a wire rack to cool. Set 18 balls aside; freeze remaining balls in an airtight container for another use. Using a small paring knife, make a small hole in the bottom of each ball. Set aside.
Make the creme patissiere: Combine milk, heavy cream, and vanilla bean in a medium saucepan. Set pan over medium heat, and bring the milk mixture just to a boil. Remove the pan from heat, cover, and let mixture steep 10 to 12 minutes. Remove and discard vanilla bean. In a small bowl, whisk together egg yolks and sugar until light and fluffy, about 2 minutes. Add flour, and continue whisking egg mixture until smooth. Slowly pour the hot milk mixture into the egg mixture. Whisk this new mixture until it is completely smooth and free of lumps. Return new mixture to pan, and place over medium heat. Bring mixture to a boil, whisking constantly; cook 2 minutes more. Transfer the pastry cream to a bowl. Lay a piece of plastic wrap directly on top of the pastry cream to prevent a skin from forming. Refrigerate until ready to use. Place half of the pastry cream in a pastry bag fitted with an Ateco #6 round tip; reserve the remaining pastry cream for assembling the cake. Pipe pastry cream into each of the 18 reserved balls. Set the filled cream puffs aside until you are ready to dip them in the caramel.
Make the caramel: Prepare an ice-water bath, and line a rimmed baking sheet with parchment paper; set both aside. Combine sugar, 1/4 cup water, and corn syrup in a small saucepan. Set over high heat, and bring to a boil. Swirl the pan occasionally until the sugar has dissolved. Continue cooking until the syrup is golden amber. Remove pan from heat, and plunge bottom of pan in the ice bath to stop cooking. Dip the top of each of the 18 filled cream puffs in the molten caramel. Place the balls, top-sides down, on the prepared baking sheet; the caramel will harden and flatten, and this surface will become the top of the cake. Make the caramel fans: Place a sheet of parchment paper on a clean work surface. Using a small offset spatula, place a small amount of molten caramel on the parchment. Starting from the center, draw out the caramel in a few quick strokes, creating a fan. Repeat making more fans with the remaining caramel. Set fans aside to harden and dry.
Make the creme chantilly: In the bowl of an electric mixer fitted with the paddle attachment, combine heavy cream, confectioners' sugar, and vanilla. Beat on high speed until the mixture forms stiff peaks, 2 to 3 minutes. Chill until ready to use. Assemble the cake: Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper, and set aside. Roll the pate feuilletee dough out to an 1/8-inch thickness. Using a plate or a bowl as a guide, cut out 2 (8-inch) circles from the dough. Prick the circles all over with a fork or a pastry docker to prevent the dough from rising. Place the circles on the prepared baking sheets. Brush the outer edge of the circles with the beaten egg. Place the reserved pate a choux dough in a pastry bag fitted with an Ateco #9824 tip. Pipe dough around the rim of each circle. With remaining dough, pipe a spiral in the center of each circle, starting in the middle. Bake until the pastry is crisp and deep golden brown, 50 to 55 minutes. Transfer baking sheets to a wire rack to cool. Dip uncoated sides of cream puffs, 1 by 1, in the caramel. Arrange on the pastries, 9 per circle, leaving space in between; the caramel will hold the cream puffs to the pastry. Combine the remaining pastry cream and 1/4 of the creme chantilly in a bowl. Place the mixed cream in a pastry bag fitted with an Ateco #9824 tip. Fill the center of each circle with cream mixture. Place the remaining creme chantilly in a pastry bag fitted with an Ateco #9824 tip. Pipe large rosettes between the caramel-covered cream puffs. Cover the center of each circle with the remaining cream. Garnish with caramel fans, and serve.
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