Prepare the Tangzhong:
In a small saucepan, whisk together the water and flour over medium heat until the mixture thickens and reaches 150°F (65°C). Remove from heat and let it cool to room temperature.

Make the Dough:
In a large mixing bowl, combine the all-purpose flour, granulated sugar, instant yeast, and fine sea salt. Add the whole milk, egg, vanilla extract, cooled Tangzhong, and softened butter. Mix until a dough forms, then knead for about 8–10 minutes until smooth and elastic.

First Rise:
Place the dough in a lightly oiled bowl, cover with a damp cloth, and let it rise in a warm place for 1–1.5 hours, or until doubled in size.

Prepare the Filling:
In a small bowl, mix together the softened butter, brown sugar, ground cinnamon, and a pinch of salt until smooth and well combined.

Shape the Cake:
Once the dough has risen, punch it down and roll it out on a floured surface into a 12×18-inch rectangle. Spread the cinnamon sugar filling evenly over the dough. Starting from one edge, carefully roll the dough into a log.

Cut and Arrange:
Cut the rolled dough into 12 equal pieces. Arrange them in a greased 9×13-inch baking pan, placing each piece cut-side up. Cover and let it rise for an additional 30–45 minutes.

Bake:
Preheat the oven to 350°F (175°C). Bake the cake for 30–35 minutes, or until golden brown on top.

Prepare the Icing:
While the cake is baking, beat together the softened butter, cream cheese, powdered sugar, milk or heavy cream, vanilla extract, and a pinch of salt until smooth and creamy.

Finish:
Once the cake is done baking, remove it from the oven and let it cool slightly. Drizzle the cream cheese icing over the warm cake before serving.