Make the tangzhong: Combine water, milk and flour in a medium saucepan. Whisk until combined and no lumps remain. Place saucepan over medium heat and cook mixture, whisking constantly, until it thickens, about 1-2 minutes. Set aside to cool.
Make the dough: While the tangzhong is cooling, begin preparing the rest of the ingredients for the dough. If you haven't already, melt the butter and warm the milk to give them a chance to cool slightly. 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, scraping down the sides of the bowl as necessary.
Increase mixer speed to medium and knead mixture for 4 minutes or until dough is smooth and clears the side of the bowl. It’s okay if it’s sticking to the bottom of the bowl. 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 an hour or until 1 ½ or 2 times its original size. The time will largely depend on the temperature of your kitchen.
In the meantime, make the filling. Combine butter, flour, salt, brown sugar and cinnamon in a medium bowl. Using a fork, mix ingredients together until a smooth, spreadable paste forms.
Once the dough has risen, preheat oven to 425 degrees. Line two baking sheets with parchment paper. Remove the dough from the bowl and place it onto a clean countertop. Roll out with a rolling pin until it is in the shape of a rectangle that is about 16 by 20 inches in size. The dough should be smooth enough that you do not need to add flour during the rolling process. If yours is sticking the rolling pin, however, sprinkle a little flour over the dough as necessary to prevent this from happening.
Using an off-set spatula, spread filling in a thin layer over the entire surface of the dough.
Fold the dough like a business letter going the long way: bring the top third of the dough down over the center third and bring the bottom third up over both the center and top third. You may need a metal dough scraper to help coax your dough from the counter top if it is sticking. Roll out folded dough just to flatten slightly.
Using your dough scraper, sharp knife or a pizza wheel, cut the folded dough into 12 equal size pieces.
Working with one piece of dough at a time, twist the strip of dough as you hold it between your thumbs and forefingers until it has a nice spiralized appearance.
While you continue to hold the dough in the thumb and forefinger of one hand, use your free hand to wrap the twisted strip of dough around the three middle fingers of the hand that holds the dough strip, tucking the last bit of the dough strip up and through the center of the circle you have just created (see photo below). Place shaped bun on a prepared baking sheet. Once you have six buns shaped and placed on one of the baking sheets, cover with plastic wrap and set aside. Continue with remaining buns, shaping and placing them on the second baking sheet before covering with plastic wrap. Let covered buns rise for an additional 30-45 minutes.
Meanwhile, make the egg wash. Whisk egg and water together in a small bowl. Once the buns have risen and are ready to go into the oven, remove plastic wrap and brush the surface of each bun with the egg wash Generously sprinkle pearl sugar over the top of each bun.
Starting with the batch of buns you shaped first, bake one pan of buns at a time for 12-14 minutes or until golden brown. Repeat with remaining pan. Move to a wire rack to cool slightly and serve. Store at room temperature in an airtight container for up to two days. Rewarm for a few minutes in a 300 degree oven if desired.