While the tangzhong is cooling, begin preparing the rest of the ingredients for the dough. Combine yeast, cardamom, salt, sugar and flour in the bowl of a stand mixer. Whisk briefly by hand to combine.
Attach the dough hook to the mixer. Add melted and cooled butter, slightly warm milk, egg and the cooled tangzhong (it can be slightly warm to the touch) to the dry ingredients. Mix on medium low until fully combined.
Increase mixer speed to medium and knead mixture for 4-5 minutes or until dough is slightly tacky but not sticky and clears the side of the bowl. It’s okay if it’s sticking to the bottom of the bowl slightly. You can also mix and knead the dough by hand but it will take longer, more like 8-10 minutes.
Cover bowl with plastic wrap and allow the dough to rise for a 1 ½ to 2 hours or until 1 ½ or 2 times its original size. The time will largely depend on the temperature of your kitchen.
Once dough has risen, line a baking sheet with parchment paper and divide the dough into 12 equal pieces. Working with one piece at a time, gather up the edges of the dough into a rough ball shape, pinching the dough where the edges come together to create a “bellybutton”. Place the ball, bellybutton side down, on a clean kitchen counter. Cupping the ball with your hand so that your palm is lightly resting on the dough and your fingertips are resting on the counter, make small circles with the dough until a tight ball is formed. Place the ball on the baking sheet. Repeat with remaining pieces of dough.
Cover lightly with plastic wrap and let rise at room temperature for 45 minutes.
Meanwhile, preheat the oven to 400 degrees. Combine egg and 1 tablespoon water to create an egg wash. Uncover the buns and brush each bun with the egg wash. Bake at 400 degrees for 12-16 minutes or until buns are golden brown. Cover the buns with a clean dish towel and let cool to room temperature (this will keep the buns from forming a hard crust).