To make the dough: In the bowl of a stand mixer fitted with the dough hook, add the yeast, ½ teaspoon cane sugar, and warm water. Stir to combine and let sit for 3 to 5 minutes or until bubbly.
In a medium pot over medium heat, add the butter and milk. Stir until completely melted, remove from heat.
Add the flour, remaining 57 grams cane sugar, and salt to the mixing bowl in that order. Add the egg and milk mixture and knead on speed 2 to 4 for 5 minutes.
Place the dough in a greased bowl, and cover the bowl with plastic wrap and let rise for 1 to 3 hours or until doubled in size.
Preheat the oven to 350°F.
To make the filling: In a bowl, add the brown sugar and cinnamon and mix to combine completely.
On a floured surface, roll out the dough to a rectangle that is about 12 x 16-inches. Using a butter knife or spatula spread the butter and cinnamon mixture all over the rolled out dough.
Starting at the larger side, fold the dough over and start to roll the dough until it is completely rolled into a log. Trim off each of the ends, then cut the dough log into 8 cinnamon rolls. Place in a greased baking dish.
Bake for 40 minutes or until golden brown.
To make the icing: In a mixing bowl, add the powdered sugar and milk and whisk until you have a smooth glaze. Pour the glaze over the top of the warm cinnamon rolls, about 10 minutes after coming out of the oven.
Store in the fridge for up to 3 days.