For the buns, stir together the milk and yeast in a medium bowl and allow to rest for 10 minutes. Meanwhile, in a large bowl, stir together the flour, brown sugar, cardamom, and salt until combined.
In the bowl of a stand mixer fitted with the dough hook attachment, combined the flour mixture and the yeast mixture and mix at medium low speed until the mixture begins to come together. Add the butter, 1 tablespoon at a time, until completely incorporated. Increase the speed to medium high and allow to mix for about 4 minutes, or until the dough is relatively smooth.
Transfer the dough to a lightly greased bowl and cover. Allow to rise at room temperature out of direct sunlight for 1 hour.
For the filling, combine the butter, brown sugar, cardamom seeds, and vanilla extract in a food processor or blender until completely smooth.
Line two baking sheets with parchment paper and set them aside.
On a lightly floured surface, roll out the dough into a roughly 14-inch by 21-inch rectangle. Use a spatula to spread the cardamom filling over the rectangle, leaving about 1/2-inch clean around the edges. Sprinkle the flake sea salt over the filling. Use a ruler or a tape measure to mark every 7 inches along the 21-inch long side of the rectangle, so you have 3 7-inch sections. Fold the right side so that the edge lines up with the farthest away 7-inch mark and the fold is on the nearest 7-inch mark. Now fold the left side over the right.
Move the rectangle so that the open sides of the rectangle are on your left and right, and start rolling the dough out again so that it is about 15 inches along the longest edge. Use a ruler or tape measure and a pastry wheel or pizza cutter to mark every 3/4-inch along the 15-inch side. Use your pastry wheel or pizza cutter to cut along the mark so you have a long strip (in the end you’ll have 20 strips total, but I recommend going one at a time).
Hold one end of the strip in your right hand, pinching it between your pointer finger, middle finger, and thumb. Hold the other end with your left hand and pull it gently to stretch it a bit and start twisting it to create a twist in the strip. Then start wrapping it around the tips of the three fingers on your right hand holding the other end of the dough. You can wrap it around your fingers 2 to 4 times depending on the size of your fingers and the length of the strip. Once you get towards the end, tuck the end of the strip through the front of the roll and then place the twisted roll on the parchment paper-lined baking sheet. You can watch the cinemagraph to help you understand. Repeat until all the strips of dough have been made into buns.
Cover the buns lightly with a kitchen towel and let them proof for 40 minutes. Preheat the oven to 425 degrees Fahrenheit.
Meanwhile, you can prepare the syrup. Heat the brown sugar, water, vanilla bean paste, and cardamom seeds in a small saucepan over medium heat. Bring the mixture to a boil, then reduce heat to low and simmer for 5 minutes. Remove from heat and set aside to cool to room temperature.
Once the buns have proofed and the oven is heated, place the pans in the oven and bake until the buns are golden, about 8 to 10 minutes. As soon as you remove the pans from the oven, use a pastry brush to brush the buns with the syrup. Once the buns are coated, sprinkle them with the flake sea salt. Allow to cool for 10 minutes until serving.
Adapted from Fix Feast Flair