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.