In the bowl of a stand mixer fitted with the paddle attachment, combine 2 cups (250 grams) flour, brown sugar, salt, and yeast.
In a small saucepan, heat milk, ½ cup (120 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 2¼ cups (281 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 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 45 minutes.
Place a large piece of parchment paper on work surface; dust with flour. Set aside.
On a very lightly floured surface, divide dough into 12 portions (about 81 grams each). Shape each portion into a ball. Roll each ball into a 20-inch rope, tapering ends. Shape each rope into a “U” shape, and cross ends of rope about 3 inches down. Twist ends once; fold down, and place ends on dough, pinching to seal. (If there is too much flour for ends to stick, place a small amount of water on dough to help seal). Place on prepared parchment paper. Cover and let rise in a warm, draft-free place (75°F/24°C) for 20 minutes.
Preheat oven to 375°F (190°C). Line 2 baking sheets with parchment paper.
In a 4-quart stockpot, bring 8 cups (1,920 grams) water and baking soda to a gentle boil.
Carefully drop pretzels, 1 to 2 at a time, top side down, into boiling water mixture. Cook for 30 seconds; turn, and cook for 30 seconds. Using a spider strainer, lift, and let excess water drip off; place on prepared pans.
In a small bowl, whisk together egg yolk and remaining 1 tablespoon (15 grams) water. Brush egg wash onto dough.
Bake until deep golden brown, 14 to 16 minutes. Let cool on pans for 5 minutes.
In a large bowl, combine granulated sugar and cinnamon.
Brush melted butter onto pretzels. Place 1 pretzel in cinnamon sugar, and toss until completely covered. Shake off excess, and place on serving platter. Repeat with remaining pretzels. Serve warm or at room temperature.