Make the dough: In the bowl of a stand mixer fitted with the paddle attachment, combine 2 cups (254 grams) flour, brown sugar, kosher salt, and yeast.
In a small saucepan, heat water and butter over medium heat until an instant-read thermometer registers 120°F (49°C) to 130°F (54°C). Add warm butter mixture to flour mixture, and beat at medium speed until combined. Reduce mixer speed to low, and gradually add 2½ cups (318 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, 5 to 6 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining ¾ cup (95g) flour, 1 tablespoon (8g) at a time, if dough is too sticky. Turn out onto a lightly floured surface, and shape into a smooth round.
Lightly oil a large 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 45 minutes.
Place a large piece of parchment on work surface, and dust with flour.
Water bath: In a 4-quart stockpot, bring water and baking soda to a slow boil over medium to medium-high heat.
Punch down dough, and let stand for 10 to 15 minutes. Turn out onto a very lightly floured surface, and divide into 12 portions (85-90g each). Cover and let stand for 10 minutes.
Roll each portion into a 22-inch-long rope that tapers at each end. (Lightly flour dough if sticking to surface.) Shape each rope into a “U,” and cross ends of rope about 3 inches down. Twist twice; fold down, and place ends on dough, pinching to seal. (If there is too much flour for the ends to stick, place a small amount of water on dough to help seal). See video below for shaping tutorial. Place on prepared parchment paper. Cover and let rise in a warm, draft-free place (75°C/24°C) until puffed, 15 to 20 minutes.
Preheat oven to 400°F (200°C). Line 2 baking sheets with silicone baking mats or parchment paper; if using parchment, spray parchment with cooking spray.
Carefully drop pretzels, 1 to 2 at a time, top side down, into boiling water mixture. Cook for 30 seconds per side. Using a spider strainer or large spatula, lift, and let excess water drip off; place on prepared pans.
Topping: In a small bowl, whisk together egg yolk and water. Brush egg wash onto each pretzel dough. Sprinkle with pretzel salt.
Bake until deep golden brown, 14 to 16 minutes. Let cool on pans for 5 minutes. Remove from pans, and place on wire racks. Serve warm or at room temperature.