How to Make It
Place a large pizza stone on oven rack in lower third of oven. Preheat oven to 425°F. Stir together milk, butter, 3/4 cup water, and salt in a medium stainless steel saucepan over medium-high. Bring to a boil, stirring occasionally. As soon as mixture boils and butter is melted, remove from heat, and add flour all at once. Stir vigorously with a wooden spoon until flour is incorporated, about 30 seconds. Return pan to medium-high heat, and cook, stirring vigorously with wooden spoon, until a thin film of flour forms on bottom of pan, 3 to 4 minutes.
Immediately transfer dough to a stand mixer fitted with the paddle attachment. Beat on medium-high speed 30 seconds to release steam. Decrease speed to medium. With mixer running, add 6 eggs, 1 at a time. Once mixture becomes cohesive, increase speed to medium-high, and beat until batter is stiff enough to hold a soft peak, about 12 minutes. (Stop mixer and lift paddle attachment. Lifted batter should form a small peak and slowly fold over itself.) Decrease speed to medium-low, and add mustard seeds, dry mustard, black pepper, and 1 cup cheese; beat until well combined.
Transfer batter to a large piping bag, and cut a 1-inch hole in the tip. Squeeze a 1 1/2-inch mound of batter (about 2 tablespoons) onto a parchment paper–lined baking sheet. Squeeze a second identical mound of batter directly on top of the first. Repeat once more to form a stack of 3 mounds of batter (like building a snowman, but with equal-size balls). Repeat piping technique to form 8 stacks of batter, 4 per baking sheet, leaving at least 3 inches between stacks. Whisk together remaining 1 egg and remaining 1 teaspoon water in a small bowl. Using a pastry brush, brush stacks with egg wash starting at the base and brushing up the sides to the top. Repeat motion, working around sides of each stack, until completely covered in egg wash and sides are smooth. Top each mound with about 1 1/2 tablespoons cheese.
Place baking sheets side by side on top of hot pizza stone, and bake gougères in preheated oven until bottom edges are just starting to lightly brown (do not open oven), about 10 minutes. Reduce oven temperature to 375°F, and bake until deep golden brown, puffed, and shell is crisp, 32 to 38 minutes. Remove gougères from oven, and, using a thin wooden skewer, immediately poke each gougère in 3 places, from the top straight down to base, to allow steam to escape. Serve warm, or let cool completely on pans, about 30 minutes.