In a large mixing bowl, dissolve yeast in 1/4 cup of warm (not more than about 100F) water and 1 tablespoon of the sugar. Let sit.
In a small saucepan, warm butter and milk to about 100F. It may help to melt the butter first and then add the milk if both are straight from the fridge.
Using a stand mixer with a dough hook combine dissolved yeast and the butter mixture on low. Slowly the sugar, then the egg. Add salt and flour, mix until smooth. The mixture should become a sticky dough at this time. Continue to mix with the dough hook and add additional flour a little at a time as needed for it become a tacky but smooth and uniform ball of dough.
Place in a greased bowl. Cover with damp cloth and let rise for about 20 minutes. Replace the cover and refrigerate overnight.
The next day:
Butter a 8x8 inch baking dish. Set aside.
Remove dough from bowl and place on a floured counter. Roll out in a 12 by 14 inch rectangle. The dough should be about an inch thick.
Using a spatula, spread the soft, creamed butter over dough. Sprinkle the sugar and cinnamon evenly over the buttered dough.
Roll dough into a tight log. Cut into 9 1 inch thick slices. Place in the prepared pan.
Allow to rise about an hour until they have almost doubled and start to drift together. Bake for about 35 minutes or until fully cooked and browned.
While you are waiting, whisk together all of the icing ingredients until a smooth, pourable icing forms—add more milk or cream if needed.
Move the pan to a wire rack. Pour the glaze over the rolls and smooth it out with a spatula.
Cool in pan for about 5 minutes prior to serving.