Brush the inside of two 8" or 9" round pans or a 9" x 13" tray with melted butter.
Stand mixer (or by hand) instructions: place the water, yeast and ½ teaspoon of sugar in a large mixing bowl and set aside. Combine the milk, ⅓ cup (66 grams) sugar, melted butter, salt and egg in another bowl or measuring jug. Stir this mixture well, then add to yeast and water in bowl.
Add half the flour and beat until smooth, using the dough hook. Add the remaining flour until the dough is firm, but will be slightly sticky. Cover, and let rise until doubled in size.
Punch down dough after it has risen and let rest for 5 minutes.
Meanwhile, mix the cinnamon with the brown and white sugar in a small bowl.
Roll out the dough with rolling pin into a 7" x 10" rectangle. Spread the melted butter on the dough, then sprinkle with the cinnamon sugar mixture, going almost to the edges. Roll the dough, starting from the longest edge (see below.) Pinch the edges to seal.
Cut into quarters using plain dental floss, then cut each quarter into 3 equal pieces. Place into two 8" or 9" round buttered tins or a rectangular baking tray, and place in oven (unheated) for about half an hour or until doubled in size.
Once risen, remove from the oven and preheat oven to 350ºF.
When ready, place the rolls in the center of the oven and bake for 20 to 25 minutes.
While the rolls are baking, make the icing.
Mix all the icing ingredients together, until it has a smooth spreading consistency.
The rolls will rise and be golden brown when ready. While still a little warm, spread with the icing.