Directions

1. In a bowl warm 3/4 cup plus 2 tablespoons milk in the microwave. Add to the bowl of a stand mixer. Mix 1 tablespoon sugar and yeast into the bowl of milk. Let sit until foamy, 8–10 minutes.
   
2. In the bowl with milk mixture add 1 egg, 6 tablespoons butter (room temperature and cut into pieces), 2 teaspoons salt, 2 teaspoons cardamom and 3 tablespoons sugar. Fit mixer with a dough hook then add flour and mix on low speed until combined, about 1 minute.
   
3. Increase speed to medium-high and mix, scraping down sides of bowl a couple of times, until dough is smooth and elastic and cleanly pulls away from sides of bowl, 12–14 minutes. Lightly coat a medium bowl with nonstick oil spray and place dough in bowl. Cover and chill at least 2 hours or up to 1 day.
   
4. Take dough out of the fridge. Mix 1/2 cup sugar, 4 teaspoons ground cardamom, and 1/2 teaspoon salt in a small bowl to combine. Set cardamom sugar aside.
   
5. Roll out dough on a lightly floured surface to an 18x11" rectangle, occasionally lifting dough and flouring surface as needed to prevent sticking. Turn so the long side is facing you.
   
6. Brush 3 butter, melted, slightly cooled, over surface of dough, leaving a 1/2" border on all sides. Evenly sprinkle cardamom sugar over melted butter. Gently roll over sugar to press into dough. Starting on one of the long sides, tightly roll up dough into a log. Pinch seam closed with your fingers to seal; transfer to a parchment-lined baking sheet and turn log seam side down. Chill 20 minutes.
   
7. Using a sharp knife, slice log lengthwise into two strands. Lay one strand over the other, cut sides up, to form an X. Braid strands together until whole length is braided, then carefully join ends together to form a ring. Using your fingers, pinch ends together as best as you can to seal. Cover loosely and let sit in a warm, draft-free spot until it’s grown about 1½ times in size and dough springs back very slowly when poked with your finger, 1½–2½ hours.
   
8. Place a rack in middle of oven; preheat to 425°. Whisk 1 large egg and 1 teaspoon water in a small bowl to combine. In a small bowl add 1 teaspoon cardamom and mix in 1 tablespoon sugar. Brush bun with egg wash and sprinkle cardamom sugar evenly all over.
   
9. Bake bun for 5 minutes. Reduce oven temperature to 350° and continue baking bun until deep golden brown, 60 minutes more. Tent the bread at 30 minutes with a piece of foil.
   
10. Meanwhile, bring brown sugar, pinch of salt, and 2 tablespoons water in a small saucepan over medium heat and cook until slightly thickened, about 1 minute. Remove syrup from heat and stir in vanilla extract. Remove bun from oven and immediately brush syrup over; transfer to a wire rack and let cool at least 10 minutes before serving.