To make the Tangzhong, whisk together the bread flour, warm water and warm milk in a small saucepan over medium heat. Stir constantly until the mixture thickens into a paste-like consistency. Remove from heat and set aside to cool.
In a small bowl, combine the instant yeast with the warm water and a pinch of sugar. Stir and let it sit for 5-10 minutes until foamy.
In a large bowl, whisk together the flour, sugar and salt. In another bowl, combine the prepared Tangzhong, yeast mixture, warm milk, softened butter and egg. Gradually add 2 cups of the flour mixture, stirring with a wooden spoon or on low speed in a stand mixer fitted with the dough hook attachment. Slowly add the remaining cups of the flour, ¼ cup at a time, until a soft, elastic dough forms. Avoid adding too much flour, as it can make the rolls dense.
Transfer the dough to a lightly greased bowl, turning to coat dough. Cover with plastic wrap or a clean dish towel and let the dough rise in a warm place for 1 hour (it will double in size).
In a small bowl, mix together the brown sugar and cinnamon. After the first rise, place the dough on a lightly floured surface. Roll the dough into a rectangle about 14 inches long and 8 inches wide and ¼ inch thick. Spread the softened butter evenly over the top of the dough, then sprinkle with the brown sugar and cinnamon mixture.
Starting at the long end of the rectangle, roll the dough into a log. Use unflavored dental floss, a sharp knife or a serrated knife to cut the log into 12 even inch slices. Place the slices on a lightly greased baking pan or 13 X 9 dish prepared with cooking spray. Cover the pan with plastic wrap or aluminum foil, and allow the cinnamon rolls to rise a second time for 60-75 minutes.
Preheat your oven to 375°F halfway through the second rise. Bake the rolls for 25 – 28 minutes or until they are golden brown. Check the internal temperature with an instant-read thermometer to ensure it's around 195°F. Remove the pan from the oven then carefully remove the rolls from the pan and place them on a wire rack to cool.
Whisk the frosting ingredients together in a small bowl. While the rolls are still warm, spread the frosting generously over the top of the rolls. The frosting will melt slightly, creating a glaze that complements the gooey cinnamon rolls perfectly.