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1
Preheat the oven to 425 degrees F. Line a large baking sheet with parchment paper and set aside.
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2
Combine the milk, butter, sugar, and salt in a medium saucepan and bring to a slow boil over medium heat and cook until the butter is melted.
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3
Remove from the heat and add the flour all at once, stirring with a heavy spoon.
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4
Return the pan to medium-low heat and stir briskly until the mixture becomes a smooth paste and pulls away from the sides of the pan in a ball, 1 to 2 minutes.
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5
Remove from the heat and let cool slightly, 1 to 2 minutes.
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6
Stirring constantly with a heavy wooden spoon, add 4 of the eggs, 1 at a time, beating well after the addition of each, and mix until smooth.
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7
Let the dough cool.
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8
Beat the remaining egg with the water in a small bowl to make an egg wash.
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9
Transfer the dough to a pastry bag fitted with a plain 3/4-inch tip and pipe out 16 ping-pong-ball-sized rounds onto the prepared sheet, about 2 inches apart.
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10
(Alternatively, spoon onto the baking sheet with a large spoon.)
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11
Lightly brush the egg wash on the tops and sides of the dough using a pastry brush and bake until doubled in size, 14 to 15 minutes.
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12
Reduce the temperature to 375 degrees F and bake until the balls are golden brown, firm to the touch, and hollow inside, 6 to 7 minutes.
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13
Turn off the oven and leave the oven door ajar for 10 minutes for the puffs to dry.
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14
Cool completely on a wire rack.
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15
Cut each profiterole in half and scoop out the doughy centers with a small spoon.
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16
To serve, fill the bottom halves with a small scoop (about 1-ounce) of the ice cream and cover with the tops.
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17
Place 2 to 4 profiteroles on each dessert plate and sift the confectioners' sugar over the profiteroles and decoratively onto the plates.
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18
Drizzle each profiterole with warm chocolate sauce and serve immediately.
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19
Combine the cream, milk, sugar, and vanilla seeds and bean in a medium heavy saucepan and bring to a gentle boil over medium heat.
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20
Remove from the heat.
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21
Beat the egg yolks in a medium bowl until frothy and lemon colored, about 2 minutes.
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22
Whisk 1 cup of the hot cream into the egg yolks in a slow, steady stream.
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23
Gradually add the egg mixture to the hot cream, whisking constantly.
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24
Cook over medium-low heat, stirring, occasionally, until the mixture thickens enough to coat the back of a spoon, about 5 minutes.
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25
Remove from the heat and strain through a fine-mesh strainer into a clean bowl.
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26
Cover with plastic wrap, pressing it directly against the surface of the mixture to keep a skin from forming.
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27
Refrigerate until well chilled, about 2 hours.
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28
Pour mixture into an ice cream maker and freeze according to the manufacturer's instructions.
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29
Transfer to an airtight container and freeze until ready to serve.
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30
Scald the half-and-half and butter in a small heavy saucepan over medium heat.
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31
Remove from the heat.
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32
Place the chocolate and vanilla in a medium heatproof bowl.
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33
Add the hot half-and-half and let sit for 2 minutes, then whisk until smooth.
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34
Serve slightly warm.
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35
(The sauce can be kept refrigerated in an airtight container for up to 3 days; rewarm gently before serving.)