Cheddar och ört Gougères

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1. 1
   Place racks in upper and lower thirds of oven; preheat to 425°. Heat ¾ cup (1½ sticks) unsalted butter, ¾ cup whole milk, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ¾ cup water in a medium saucepan over medium-high, whisking occasionally to prevent scorching, until butter is melted and mixture is boiling (it should bubble vigorously, but watch it closely so it doesn’t boil over). Immediately remove pan from heat and add 1½ cups (188 g) all-purpose flour all at once. Whisk until incorporated and a paste forms. Return to medium-high heat and cook, mixing vigorously with a wooden spoon, until dough is the consistency of smooth mashed potatoes, about 30 seconds. Continue cooking, stirring constantly, until dough starts to dry out, creates a film on bottom of pot, and releases a lot of steam, about 3 minutes.
2. 2
   Transfer dough to the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed 1 minute. Reduce speed to medium-low and add 5 large eggs one at a time, beating to combine after each addition. Scrape down sides of bowl with a rubber spatula. Beat in 1 large egg white. Increase speed to medium and beat until dough holds medium-soft peaks when you lift paddle out of dough, 8–10 minutes. (The dough should be at or just above room temperature and texture should be visibly voluptuous, glossy, and very smooth. When you pinch a bit between your thumb and index finger, you should be able to bring your fingers apart and have the dough stay connected by a thread.)
3. 3
   Using a rubber spatula, fold 3 scallions, thinly sliced, 4 oz. sharp yellow cheddar, grated, 2 Tbsp. finely chopped dill, 1 Tbsp. whole grain mustard, and ¼ tsp. freshly ground pepper into dough.
4. 4
   Beat 1 large egg yolk and 1 tsp. water in a small bowl to blend; set egg wash aside.
5. 5
   Scrape dough into a pastry bag fitted with a round ½" tip or a plastic bag with 1 corner cut on a diagonal to make a ½" opening. Pipe 1"-diameter rounds (about the size of a heaping tablespoon) between 2 parchment-lined baking sheets, spacing about 1" apart. (Piping rows of 5 down short sides and 6 across the long sides works well.) Brush tops with reserved egg wash and sprinkle remaining 4 oz. sharp yellow cheddar, grated, over.
6. 6
   Bake gougères 10 minutes, then reduce oven temperature to 350° (do not open oven) and bake 15 minutes. Rotate baking sheets top to bottom and front to back, then continue to bake until golden brown and firm to the touch, 5–7 minutes more. Serve warm or let cool. Do ahead: Gougères can be piped 1 week ahead. Freeze on baking sheets until solid, then transfer to resealable plastic bags and freeze. Brush with egg wash and top with cheese just before baking. Bake from frozen, adding about 3 minutes to final bake time. Gougères can be baked 2 days ahead. Let cool, then store airtight at room temperature, or freeze up to 1 month. Reheat in a 250° oven until warm to the touch, about 5 minutes, or 8 minutes if frozen.