Special equipment: a large pastry bag fitted a large plain tip, an electric mixer fitted with a paddle attachment, a pastry brush
Preheat the oven to 425 degrees F.
Put the water, butter, and salt in a heavy 2-quart saucepan. Bring to a boil over high heat, stirring occasionally, to melt the butter. Remove the pan from the heat and add the flour all at once. Stir with a wooden spoon until the dough comes together to form a ball. Place the pan over medium heat and beat constantly until the dough pulls away from the sides of the pan, about 4 minutes. Do not let it burn. A film may form on the bottom of the pan.
Put the dough in an electric mixer fitted with a paddle attachment. Add the eggs 1 at a time, beating well after each addition. Scrape down the sides of the bowl and continue to beat the dough for 4 minutes at high speed. Feel the dough: it should be sticky at first and then smooth and glossy.
Line a sheet pan with parchment paper, and use a little bit of the dough to stick the corners of the paper down. Transfer the dough to a piping bag fitted with a large tip. Pipe the dough into 2-inch balls spaced about 1-inch apart on the parchment paper. Bake for 15 minutes; then rotate the tray and lower the oven temperature to 350 degrees F. Continue to bake for 10 minutes. Turn the oven off, leaving the door slightly open to let the profiteroles cool in the oven.
To assemble, cut the profiteroles in 1/2 using a serrated knife. Place a scoop of ice cream on the botton 1/2 of the profiterole. Put the top on and drizzle with caramel sauce.
Put the sugar and water in a 2-quart saucepan over medium heat. Stir once or twice to dissolve the sugar. Increase the heat to high and use a pastry brush dipped in cold water to brush down any crystallized sugar that forms on the side of the pan. Let the mixture bubble and caramelize until it is amber, about 5 to 7 minutes. Remove from the heat and whisk the cream in very slowly. Stir in the butter. Cool to room temperature before serving.