For the profiteroles: Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Combine 1 cup water, the butter, salt and sugar
in a 4-quart pot and bring to a boil. Gradually add the flour while stirring constantly. Stir the mixture vigorously over low heat until it forms a loose dough. Be careful not to scorch the bottom of the pot. Remove from the heat and stir briefly to dry out, returning to the heat for 2 minutes more. Remove from the heat and transfer to a bowl.
Add 4 of the eggs one at a time and stir, waiting until each egg is fully incorporated before adding the next. Test the dough to see if the texture is correct: the dough should be suspended in a thread between two fingers when pinched and separated. (If this doesn't happen, add another egg.)
Transfer the mixture to a piping bag fitted with a straight pastry tip. Pipe out quarter-size rounds onto the baking sheet, spaced about 1-inch apart.
Wet your fingertips with water and press down the tips of the rounds to flatten them. Make an egg wash
by beating the remaining egg with a fork in a small bowl. Gently brush the egg wash over the tops of the dough rounds.
Bake until the dough
has puffed up and the tops are nicely browned, 8 to 12 minutes. Lower the heat to 300 degrees F and let sit in the oven for another 5 minutes. Transfer to a rack to cool. A well-cooked profiterole
should be hollow on the inside and still a little soft inside. Let cool completely.
Whip the heavy cream to stiff peaks in a large bowl. Fit a second pastry bag
with a straight or star tip and fill with the whipped cream
. Cut the profiteroles in half horizontally and fill with the whipped cream. Top with the lid. Set aside in the refrigerator until ready to serve.
For the chocolate sauce: Heat the cream in a pot until just scalded. Pour the cream over the chopped chocolate. Tap the bowl to allow the cream to settle. Wait 2 minutes, and then stir vigorously with a whisk
until smooth and emulsified. Set aside to cool or until ready to serve. Serve about 1/2 cup of the sauce with the profiteroles.