Preheat the oven to 425°F. Lightly grease (or line with parchment) two baking sheets., To make the choux pastry: Combine the water, butter, and salt in a medium-sized saucepan. Heat until the butter has melted, and bring to a rolling boil., Remove from the heat and add the flour all at once, stirring vigorously until the mixture smooths out and comes together into a ball; this should take less than 30 seconds. If needed, return the pan to the heat for a few minutes to help the batter come together., Remove the pan from the heat, and transfer the dough to the bowl of a stand mixer. After 7 minutes, check the temperature of the dough: you should be able to hold a finger in it for a few seconds. If you have a digital thermometer, the temperature should be below 125°F., Beat in the eggs one at a time; the mixture will look curdled at first, but when you add the last egg it should become smooth. Beat for 1 minute after adding the last egg. You'll have a stiff, smooth batter., Transfer the batter to a pastry bag with a 1/2" diameter round tip. Pipe into mounds about 1" in diameter (the puffs will double in volume as they bake), leaving 2" of space between them. Alternately, if you don't have a pastry bag, drop the batter by teaspoonfuls onto the prepared sheets (a teaspoon cookie scoop works well here)., Bake the puffs for 15 minutes, then reduce the oven temperature to 350°F and bake for 20 to 25 minutes more, until they're a medium golden brown. Don't open the oven door while the puffs are baking., Remove the puffs from the oven, and use a sharp knife to cut a slit into the side of each for steam to escape. Return to the oven for 5 minutes, then remove from the oven and transfer to a rack to cool., To make the filling: Combine the pudding mix, milk, and vanilla, whisking or beating until the mixture starts to thicken. Refrigerate until set., When they're cool enough to handle, split the puffs in half to make tops and bottoms; doing this will expose the center to air and keep them from becoming soggy., To assemble: Fill each of the puff bottoms with a heaping tablespoon of pudding. Replace the top. It's best to do this within an hour of when you'll be serving the puffs, otherwise they get soggy. Sift some confectioners' sugar atop the puffs just before serving., For profiteroles, scoop a generous tablespoon of ice cream onto the bottoms of the puffs; replace the tops. Drizzle with sauce before serving.