Make dough: In a medium saucepan, heat milk and water over medium heat to scalding. (An instant-read thermometer should register 180°F/82°C.) Remove from heat and add butter; stir occasionally until butter is melted and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).
In the bowl of a stand mixer fitted with the paddle attachment, combine 2 cups (250 grams) flour, sugar, salt, and yeast. Add warm milk mixture and beat at low speed until combined. Add egg, beating until combined. With mixer on low speed, gradually add 2 cups (250 grams) flour, beating just until a shaggy dough comes together and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, about 24 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 dough onto a clean surface; knead 5 to 6 times, 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, about 1 hour.
Make filling: In a small bowl, whisk together cinnamon, nutmeg, cardamom, and allspice.
In a medium bowl, stir together brown sugar and room temperature butter until smooth and combined. Reserve 1 tablespoon (6 grams) cinnamon mixture; stir remaining cinnamon mixture into sugar mixture until combined.
Line 2 baking sheets with parchment paper.
Shape twists: Punch down dough and let stand for 5 minutes. Divide dough into 10 portions (about 95 grams each). On a clean surface, roll 1 portion into an 8x4-inch rectangle. Spread 1½ tablespoons (22 grams) sugar mixture onto rectangle, leaving a ⅛-inch border along one long side. Starting at long side without border, roll up dough into a log, and press edge to seal. Roll dough into a rope about 14 inches long, dusting work surface with flour as needed.
(See below for how-to photos.) Cut rope in half lengthwise to create 2 strands; cut each strand in half crosswise (A). Place the 4 strands, cut side up, in a hashtag pattern (over, under, over, under) (B). Working clockwise, cross each bottom strand over the strand next to it (C and D). Repeat crossing procedure with remaining bottom strands (E). After all 4 sides have been crossed, cross 1 new top strand under adjacent strand (F). Repeat crossing procedure with remaining new top strands (G). Tuck ends under to shape into a round, and place on prepared pan (H). Repeat procedure with remaining dough portions and remaining sugar mixture. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.
Preheat oven to 350°F (180°C).
Make egg wash: In another small bowl, whisk together egg and water. Brush egg wash onto rolls.
Bake twists: Bake until golden brown and an instant-read thermometer inserted in center registers at least 190°F (88°C), 15 to 20 minutes, rotating pans halfway through baking.
Make topping: In a small bowl, stir together reserved cinnamon mixture and granulated sugar. Brush melted butter onto hot rolls, sprinkle with spiced sugar. Let cool on pans for 10 minutes. Remove from pans and let cool on a wire rack for 20 minutes. Serve warm. Store in an airtight container for up to 2 days.