Buttery Brioche Hot Dog Buns

Stir together 1/4 cup water, yeast, and 1 teaspoon sugar in a small bowl. Let stand until foamy, about 5 minutes. 

Combine warm milk and 2 beaten eggs in the bowl of a stand mixer fitted with the dough hook attachment. Stir in yeast mixture. With mixer running on medium-low speed, gradually beat in flour and salt. Increase speed to medium, and beat until dough is very sticky, about 2 minutes. Gradually beat in remaining 1/4 cup sugar, 1 teaspoon at a time, until well incorporated, about 4 minutes. Stop mixer, and add butter. Beat on medium speed until dough is smooth, elastic, and pulls away from sides of bowl, about 8 minutes, stopping to scrape down sides of bowl if needed. Transfer dough to a bowl lightly greased with cooking spray. Cover with plastic wrap, and let rest at room temperature until doubled in volume, 1 hour to 1 hour and 30 minutes.

Preheat oven to 375°F. Turn dough out onto a clean work surface. Divide dough into 12 (3-ounce) portions. Pat 1 dough portion into a 5- x 3-inch rectangle. Fold the 2 short sides in so they meet in the middle, and gently press. Using your hands and starting at 1 long side, roll dough into a log, gently rolling and pressing into a 5-inch-long log. Repeat with remaining dough portions. Place dough logs parallel to each other, 1 inch apart, on 2 parchment paper–lined baking sheets; gently flatten tops of each. Cover buns loosely with plastic wrap, and let rise at room temperature until almost doubled in volume, about 1 hour. 

Whisk together remaining egg and remaining 1 teaspoon water. Brush buns on 1 baking sheet with half of egg wash. Sprinkle with poppy seeds, if desired. Bake in preheated oven until golden brown and internal temperature registers 190°F, 14 to 18 minutes. Transfer buns to a wire rack, and let cool completely, about 1 hour. Repeat with remaining baking sheet of buns, remaining egg wash, and, if desired, poppy seeds. Store buns at room temperature in a ziplock plastic bag for up to 3 days.