Preheat the oven to 400°F (200°C) and place a rack in the middle position. Line a baking sheet with parchment paper.

In a medium saucepan combine ¼ cup plus 2 tablespoons water, 2 tablespoons milk, 4 tablespoons unsalted butter, 1½ teaspoons sugar, and ¼ teaspoon salt. Heat over medium, stirring occasionally, until the mixture comes to a boil and the butter is fully melted.

Reduce the heat to low, add ½ cup all-purpose flour all at once, and stir vigorously with a wooden spoon until the dough comes together into a smooth ball and pulls away from the sides of the pan, about 1 minute.

Transfer the dough to the bowl of an electric mixer fitted with the paddle attachment (or a large mixing bowl if mixing by hand). Let the dough cool for 5 minutes, then beat on low speed for 30 seconds to release steam.

With the mixer on low, add the 2 large eggs one at a time, beating well after each addition and scraping down the bowl as needed. After the second egg is incorporated, continue mixing for 1 minute more, until the dough is smooth, thick, and pipeable.

Fit a piping bag with a ½-inch round tip and fill it with the choux dough. Pipe 1¾-inch mounds onto the prepared baking sheet, spacing them about 2 inches apart. Wet your fingertip and gently smooth any pointed tips.

Bake on the middle rack for 22 to 25 minutes, until the puffs are golden and well risen. Do not open the oven during baking. When the baking time is up, turn off the oven, crack the oven door open slightly, and let the puffs sit inside for 30 minutes to dry out. Transfer the puffs to a wire rack and cool completely.

Make the chocolate sauce: heat ¼ cup heavy cream in a small saucepan over medium heat until steaming (do not boil). Remove from the heat, add 2 oz finely chopped semisweet or bittersweet chocolate and a pinch salt, and stir until smooth. (Alternatively, heat the cream in a microwave-safe bowl, then stir in the chocolate.) Let the sauce cool slightly so it is warm but not scalding before drizzling.

For ice cream–filled profiteroles: on a parchment-lined baking sheet scoop small balls of vanilla ice cream (or flavor of choice) and freeze until firm, at least 30 minutes and up to 2 hours (cover if freezing longer). Once the profiteroles are completely cool, slice each one in half horizontally with a serrated knife. Place a frozen ice cream ball on the bottom half, replace the top, drizzle with warm chocolate sauce, and serve immediately.

For pastry cream–filled profiteroles: whisk chilled vanilla pastry cream until smooth. Using a small knife, make a ¼-inch hole in the bottom of each cooled profiterole. Transfer the pastry cream to a piping bag fitted with a small round tip and pipe into each puff until nearly full. Drizzle with warm chocolate sauce and serve.