8 tablespoons unsalted butter (1 stick), cut into 8 pieces
1/2 cup whole milk
1/2 cup water, plus more as needed
1 tablespoon granulated sugar
1/4 teaspoon fine salt
1 cup all-purpose flour
4 large eggs, at room temperature
Heat the oven to 425 degrees F and arrange a rack in the middle. Line a baking sheet with parchment paper; set it aside. Heat the butter, milk, measured water, sugar, and salt in a medium saucepan over medium-high heat until the butter has melted and the liquid has come to a boil. Reduce the heat to low, add the flour all at once, and stir vigorously with a wooden spoon until well incorporated. Cook, stirring constantly, until the dough looks shiny and feels smooth to the touch, and a thin film has formed on the bottom of the pan, about 5 minutes. (The dough will easily form 1 large ball.) Transfer the dough to the bowl of a stand mixer fitted with a paddle attachment and let it cool for 5 minutes. Meanwhile, fill a small bowl with water and set it aside. With the mixer on medium-low speed, beat in the eggs 1 at a time, letting each egg completely incorporate before adding the next, until the mixture is smooth, sticky, and glossy, about 4 to 5 minutes total. Transfer the dough to a large resealable bag and cut off about 3/4 inch from one corner of the bag. Pipe 12 (2-inch-wide) round mounds onto the prepared baking sheet, making sure theyre about 1 1/2 inches apart. Dip your finger in the bowl of water and smooth the top of each mound. Place the baking sheet in the oven, reduce the temperature to 350 degrees F, and bake until the puffs are golden brown, airy, and completely dry inside, about 45 to 50 minutes (do not open the oven door until at least 45 minutes have passed). Transfer the baking sheet to a wire rack and let the puffs cool completely, about 30 minutes.