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Mix 1 Tbsp. granulated sugar and 1½ tsp. active dry yeast in a small bowl to combine.
Heat ¾ cup plus 2 Tbsp. whole milk in a small saucepan over medium-low until lukewarm (110–115°). Add warm milk to bowl with yeast and sugar and whisk to combine. (Alternatively, microwave milk in a small microwave-proof bowl on medium in 15-second increments until warm, then whisk in sugar and yeast.) Let sit until foamy, 8–10 minutes. (If yeast doesn’t foam, discard and start over with new yeast.)
Place milk mixture, 1 large egg, room temperature, 6 Tbsp. unsalted butter, room temperature, cut into pieces, 2 tsp. Diamond Crystal or 1 tsp. Morton kosher salt, 2 tsp. ground cardamom, and remaining 3 Tbsp. granulated sugar to the bowl of a stand mixer fitted with the dough hook. Then add 3 cups (375 g) all-purpose flour and mix on low speed until combined, about 1 minute.
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. (You can also perform the windowpane test to determine whether the dough has built up enough gluten: Pinch off a golf ball–size piece of dough and stretch between your fingers until dough is as thin as you can make it without breaking. If you can see light through the dough, it’s ready.) Lightly coat a medium bowl with vegetable oil or nonstick vegetable oil spray and place dough in bowl. Cover and chill at least 2 hours or up to 1 day. (If you don’t have a stand mixer, you can make this dough by hand: Mix all ingredients together in a large bowl with a wooden spoon until combined, then knead dough in bowl until it starts to come together into a single mass; it will be extremely sticky. Turn out dough onto a floured surface and knead, flouring surface more as needed to prevent sticking, until dough is smooth and elastic, 12–14 minutes.)
Mix ½ cup (100 g) granulated sugar, 4 tsp. ground cardamom, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt in a small bowl to combine. Set cardamom sugar aside.
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 a long side is facing you.
Brush 3 Tbsp. unsalted butter, melted, slightly cooled, over surface of dough, leaving a ½" border on all sides. Evenly sprinkle reserved 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 (this will make the dough easier to braid).
Carefully slide dough log with parchment paper onto a cutting board. 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 (a small gap in the center of the ring is fine). Using your fingers, pinch ends together as best as you can to seal (it’s okay if it’s not perfect). Carefully slide bun with parchment paper back onto baking sheet. 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.
Place a rack in middle of oven; preheat to 425°. Whisk 1 large egg and 1 tsp. water in a small bowl to combine. Coarsely crush seeds from 20 green cardamom pods in a spice mill or with a mortar and pestle. Transfer to a small bowl (or use 1 tsp. ground cardamom) and mix in remaining 1 Tbsp. granulated sugar. Brush bun with egg wash and sprinkle coarse cardamom sugar evenly all over.
Bake bun 5 minutes. Reduce oven temperature to 350° and continue baking bun until deep golden brown and an instant-read thermometer inserted into the center registers at least 180°, 25–30 minutes more. (Some filling will leak out while bun is baking, which is normal.)
Just before bun is finished baking, bring 3 Tbsp. light brown sugar, a small pinch of kosher salt, and 2 Tbsp. water in a small saucepan over medium heat and cook until slightly thickened, about 1 minute. Remove syrup from heat and stir in ¼ tsp. vanilla extract (if using).
Remove bun from oven and immediately brush syrup over; let cool at least 10 minutes.
Serve bun warm or room temperature, slicing so everyone gets some of the crunchy caramelized cardamom sugar.