Make the dough according to the tangzhong bun dough method.
Press down the dough onto a tray, wrap well, and chill in the fridge overnight to proof.
For the spiced butter, mix the soft butter with the demerara sugar, light brown sugar, salt, cardamom, and cinnamon until well combined.
Roll out the dough to about 16x10 inches. Spread 7 tablespoons (100 grams) of the spiced butter onto the dough and perform a letterfold/single fold.
Wrap and put into the freezer for 10 minutes, or the fridge for 30 minutes, to firm up.
Turn the dough 90 degrees, then roll out the dough to about 16x10 inches again and spread with the rest of the spiced butter.
Perform another letterfold/single fold. Cut the dough into 1¼-inch wide pieces.
Braid the strips, then roll up the dough into a bun. Place into a cupcake pan, lightly cover, and proof for 1 hour or until puffy.
Preheat the oven to 400°F.
Gently brush the buns with egg wash and bake for 18–22 minutes, until golden.
For the soak, whisk together all the ingredients.
Remove the buns from the pan and place into your soaking custard.
Bake at 375°F for 10 minutes.
For the icing, heat the sugar, water, and vinegar on the stovetop for 1–2 minutes.
Whisk in the butter, rum, and salt. Let cool and thicken slightly, then drizzle the buns with the icing.
