Whisk together ¾ cup milk and 4 ½ tablespoons bread flour in a small saucepan. Heat over medium heat, stirring constantly until a paste forms, about 5 minutes. The Tangzhong is ready when a spoon leaves a line in the pan. Allow to cool slightly.

Add the cooled tangzhong and the remainder of the dough ingredients to the bread machine in the order listed. Select the dough cycle and start the machine.

Whisk together the brown sugar, bread flour, cinnamon, and salt until combined. Add melted butter; stir to thoroughly combine. Set aside.

Beat cream cheese, butter, confectioners' sugar, and vanilla extract together until smooth and creamy. Set aside.

Preheat oven to 350°F.

When the dough cycle is finished, remove the dough and roll it out to a 10×28-inch rectangle on a floured surface. Trim the edges as needed. Spread the cinnamon/sugar mixture on the dough to within 1 inch of the edges.

Starting at one of the long edges, roll the dough tightly into a 28-inch log. Pinch the edges to seal the log. Place the rolled log onto the prepared baking sheet, seam side down, and form the dough into a ring. Moisten the ends of the dough with a little water; pinch the two ends together to seal.

Brush king cake with egg wash. Using a sharp knife, slash the top of the king cake to allow for steam to escape. Let rise in a warm place for about 20 minutes.

Preheat oven to 350°F. Bake the king cake in preheated oven until the top is golden brown, about 30 minutes.

Remove the king cake from the oven, and cool for about 10 minutes on a wire rack set over a rimmed baking sheet. While the cake is still warm, spoon the icing onto the king cake, allowing it to dribble down the sides of the cake. Immediately, sprinkle purple, green and yellow food glitter on the cake.