Prepare the choux pastry. Beat the eggs. Sift the cake flour twice if you can. Set a 1.2 cm diameter round tip on a pastry bag.
Put the butter, water and salt in a pan and heat over high heat. Bring to a boil while melting the butter. When it is bubbling, put all the flour in at once.
Turn the heat down to medium and mix with a heatproof spatula. When it is no longer floury, keep mixing and kneading for about a minute. When the dough comes off the bottom of the pan cleanly in one mass, turn the heat off.
Add the beaten egg in several batches, mixing well each time. With the last batch of eggs, see how stiff the dough is and adjust the amount you add. See the next step.
When you scoop up the dough with your spatula, it should plop off it in about 3 seconds - this is the consistency to aim for. The dough left on the spatula should form a peak.
Put the choux dough in a pastry bag, and make 10 rounds about 5 cm in diameter. Make a pattern on the top with a fork dipped in egg (to make them puff up evenly).
Bake in a preheated 390F/200C oven for 10 minutes, then lower the temperature to 355F/180C and bake for another 20 minutes. Turn the oven off and leave the choux buns in the oven for another 15 minutes to dry out.
Make the ice cream while the choux buns are baking. Put the egg yolks in a bowl, add vanilla beans and beat with a handheld mixer.
Combine water and sugar in a small pan, and heat to make a sugar syrup, stopping just before it starts to caramelize. Add this syrup to the bowl with the egg yolks little by little while whipping at high speed.
Keep whipping with a handheld mixer until it becomes white and thick. You should see lines when you lift up the beaters on the surface of the mixture.
Whip the cream in another bowl until it forms stiff peaks. Combine this with the mixture from Step 10 and stir until there are no streaks.
Put the creamy mixture into a pastry bag fitted with a choux pastry tip, and fill the cooled down choux puffs by inserting the tip into the side of each puff. Freeze, and they're done.
Story Behind this Recipe
I thought parfait ice cream, which doesn't need to be mixed after freezing, would be perfect for filling choux puffs.
Make the choux puffs quickly without resting in between. Do not open the oven for at least 25 minutes after putting the choux in the oven. You can make the sugar syrup to add to the egg yolks for the ice cream easily in the microwave. Combine water and sugar in a heatproof container, and microwave for 2 to 2 and a half minutes at 600w. Make sure to wear oven mitts when taking the container out of the microwave.