Step 1

Pate a Choux: Place an 11-by-17-inch piece of parchment lengthwise on work surface. Using a pencil and ruler, draw sets of parallel lines to make a grid that will be the guide for piping out the swans: Starting 3/4 inch up from the bottom edge, draw a line across the paper. Draw a second parallel line, 2 1/2 inches above the first. Draw a third parallel line, 1 inch above the second line, and a fourth line 2 1/2 inches above the third. Draw another line 1 inch above the fourth line, and a sixth line 2 1/2 inches above that. This should leave a 3/4-inch border at the top. Place parchment, pencil-side down on an 11-by-17-inch unrimmed baking sheet. Set aside.

Step 2

Preheat oven to 450 degrees. In a medium saucepan, heat butter, granulated sugar, salt, and 1 cup water over medium; bring to a boil. Remove from heat. Using a wooden spoon, quickly stir in flour. Cook over medium-high heat, stirring constantly, until mixture pulls away from sides of pan and a film forms on bottom, about 3 minutes.

Step 3

Transfer to the bowl of an electric mixer fitted with the paddle attachment. Beat on low speed until slightly cooled, 1 1/2 to 2 minutes. With mixer on low speed, add 4 eggs, one at a time, beating until each is fully incorporated and a soft peak forms when you touch dough with your finger.

Step 4

Fit a large 18-inch pastry bag with a 1/2-inch round tip (such as Ateco #806). Fill with a generous three-quarters of the dough. Fit a small pastry bag with an 1/8-inch round tip (such as Ateco #11). Fill with remaining dough. Using the large pastry bag, squeeze out 18 large teardrops of dough onto prepared baking sheet, spacing 1 inch apart and staying between the lines. Form the heads: On a separate parchment-lined unrimmed baking sheet, using the small pastry bag, squeeze out 18 small S shapes, like question marks but with the second end slightly curved as well. Make a pointed beak by pulling up the tip at the edge of the head.

Step 5

Brush beaten egg over swan bodies and heads. Bake bodies 10 minutes, then reduce oven temperature to 375 degrees and add heads. Continue baking until golden and airy, about 25 minutes. Transfer sheets to wire racks and let cool completely, about 1 hour.

Step 6

Assembly: Gently slice off top of bodies with a sharp serrated knife. Cut tops in half. Working quickly, fill each body with a small scoop of ice cream. Place both pieces of the top back on ice cream to simulate wings; place a head into the ice cream at the point where the two wings meet. Transfer swans to a baking sheet and freeze until ready to serve.

Step 7

Place chocolate in a medium bowl. Heat cream over medium; pour over chocolate. Let stand 5 minutes; stir until smooth. Let cool at room temperature until ready to use. Spoon a pool of chocolate sauce on each plate and top with a swan. Dust with confectioners' sugar. Serve immediately.