Preparation

Make the caramel for the filling

• Fill a small cup with water and place a pastry brush in it (this will be used for washing down the sides of the pan to prevent crystallization).
• In a clean, heavy-bottomed 2-quart saucepan, stir together the sugar, water, and cream of tartar. Cook over medium-high heat, occasionally washing down the sides of the pan with the pastry brush to wash away any sugar crystals, until the mixture starts to color around the edges. Gently swirl the pan to ensure that the sugar caramelizes evenly and continue to cook until the caramel turns a medium-dark amber color. Remove the pan from the heat and carefully add 3/4 cup of the cream (the mixture will bubble up furiously). Once the bubbling has subsided, stir the mixture until it is completely smooth. If there are any hardened bits of caramel in the mixture, return the saucepan to medium heat until they have dissolved. Stir in the salt and vanilla. Pour the hot caramel mixture through a fine-mesh sieve into the bowl of an electric mixer (or a different mixing bowl if you have only one electric mixer bowl; you will need it for the choux) and let cool for 10 minutes.
• Cover the bowl with plastic wrap and refrigerate the caramel until chilled, about 2 hours. Meanwhile, make the choux.

Make the choux paste

• Preheat the oven to 400°F.
• In a medium saucepan, combine the water, milk, sugar, salt, and butter and bring to a full boil over medium-high heat, stirring frequently. Remove the pan from the heat, add the flour all at once, and stir vigorously with a wooden spoon until the flour is completely incorporated and the mixture pulls away from the side of the pan. Return the pan to the heat and continue to cook for another minute, stirring, to dry out the dough a bit. Transfer the dough to the bowl of an electric mixer fitted with the paddle attachment and let it cool for 2 minutes.
• While mixing on medium speed, add the eggs, one at a time, mixing until the first egg is completely incorporated before adding the next. Mix until you get a smooth, shiny dough that falls heavily from a spoon, about 2 minutes.

Pipe the cream puffs

• Line two baking sheets with silicone baking mats or pieces of parchment paper. Scrape the choux paste into a large pastry bag fitted with a 5/8-inch plain tip, such as Ateco #7. Pipe out 2-inch mounds of the choux paste onto one baking sheet, leaving 1-1/2 inches between each cream puff. Brush the surface of each cream puff with the egg wash. Use your finger to smooth out the points on the surface of each one. (Leave the remaining choux paste in the pastry bag until the first batch of cream puffs has been baked.) Bake the cream puffs for 15 minutes, then reduce the heat to 350°F degrees and bake until puffed up and light golden brown, 10 to 15 minutes longer. Let cool on the baking sheet. Pipe out and bake more cream puffs with the remaining batter.

Finish the filling and fill the cream puffs

• Remove the chilled caramel from the refrigerator and gradually stir in the remaining 1 cup cream with a rubber spatula until blended. Place the bowl on the mixer stand (or transfer the cream to a mixer bowl) and, using the whisk attachment, beat the filling on high speed until medium-firm peaks form. Transfer the filling to a pastry bag fitted with a 1/4-inch plain tip, such as Ateco #802.
• Use a small, plain pastry tip to make a hole in the center of the bottom of each cream puff. Gently pipe the caramel filling into the hole, filling the puff completely. Lightly dust the cream puffs with sifted confectioners’ sugar before serving. If you are not serving the cream puffs immediately, place them in an airtight container and refrigerate. The shells can be made up to 3 days in advance and stored in an airtight container at room temperature, but don’t fill them more than 3 hours before serving.