Activate the yeast: In a large bowl or measuring jug, combine the warm milk, instant yeast, and 1 tablespoon of the sugar. Stir gently and let it sit for 10 minutes, or until foamy.
Prepare the tangzhong: In a small saucepan, cook the ⅔ cup milk and ¼ cup bread flour over medium heat, stirring constantly until it forms a thick paste. Remove from the heat and allow it to cool slightly.
Add the eggs and melted butter to the activated yeast mixture and whisk until combined.
In a large mixing bowl, combine the bread flour, remaining sugar, salt, nutmeg, and ground cloves.
Add the yeast mixture, cooled tangzhong, and grated carrot to the dry ingredients. Knead for about 8–10 minutes, or until the dough is smooth and elastic.
Place the dough in a lightly greased bowl, cover, and let it proof for 1 hour, or until doubled in size.
Roll the dough into a large rectangle.
Spread the softened butter evenly over the dough, then sprinkle with the brown sugar and cinnamon.
Roll the dough tightly into a log and cut into 12 equal rolls.
Arrange the rolls in a greased baking dish, cover, and let them proof again for 45–60 minutes, or until puffy.
If desired, pour a small amount of milk or fresh cream over the rolls just before baking.
Bake in a preheated 180°C (350°F) oven for 25–30 minutes, or until lightly golden.
While the rolls are baking, beat together the cream cheese, softened butter, icing sugar, and vanilla until smooth.
Spread the cream cheese frosting over the warm cinnamon rolls and serve.
