In the bowl of a stand mixer fitted with the paddle attachment, combine 1¼ cups (156 grams) flour, brown sugar, yeast, and kosher salt.

In a small saucepan, heat milk, ⅓ cup (80 grams) water, and butter over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm milk mixture to flour mixture, and beat at medium speed until combined. Reduce mixer speed to low, and gradually add 1½ cups (188 grams) flour, beating just until combined and stopping to scrape sides of bowl.

Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, 4 to 5 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out onto a lightly floured surface, and shape into a smooth round.

Lightly oil a medium bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 40 minutes.

Preheat oven to 350°F (180°C). Spray a 9-inch square baking pan with cooking spray. Line pan with parchment paper, letting excess extend over sides of pan.

On a lightly floured surface, roll dough into an 11-inch square. Score a 9-inch square in center of dough. Brush water onto 9-inch square. Fold dough over at score mark. Using hands, lift dough, and place in prepared pan, making sure dough is even and fills corners of pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 20 minutes.

Using your fingertips, dimple center square back down, leaving outside border as is. In a small bowl, stir together 1½ tablespoons (22.5 grams) hot water and baking soda. Gently brush hot water mixture onto outside border; brush egg yolk onto outside border. Pour warm Beer Cheese Filling in center.

Bake for 25 minutes. Brush crust with egg white, and sprinkle with pretzel salt. Bake until crust is golden brown and an instant-read thermometer inserted in filling registers at least 160°F (71°C) (but not over 175°F/79°C), about 5 minutes more. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool on a wire rack. Garnish with chives and red pepper, if desired.