Dissolve yeast in the water along with 1 teaspoon sugar. Set aside to proof. In a medium saucepan, scald the milk with the butter, salt, and 1/3 cup sugar. Let cool. Mix in beaten eggs and vanilla. In a large bowl, mix 3-1/2 cups of the flour with the yeast mixture and then the milk mixture. Stir to form a raggard dough. Turn out onto a floured board and knead for about 10 minutes; adjust the flour so the dough is smooth, not too sticky.
Meanwhile, clean the bowl, dry it, and then grease it with oil or butter. Return the dough to the bowl and roll it around to coat the outside with oil. Cover well with plastic wrap. Let rise in a warm place until it doubles in bulk (usually 1 to 1-1/2 hours).
Punch down and roll out on a lightly floured board to about 1/4 inch thickness. You should end up with an 18 inch-by 16 inch rectangle. Brush well with melted butter. In a small bowl, mix the remaining 1/2 cup sugar and the cinnamon together. Sprinkle evenly over dough. Starting at the narrow end, roll to form a log. With a sharp knife, cleanly cut 12 thick slices (each a little more than an inch wide) and place on a greased baking sheet, cut-side up. (Pinch ends if necessary.) Cover and let rise again for about 30 minutes.
Preheat oven to 375 degrees F. Bake until golden brown, about 15 to 20 minutes. Meanwhile, prepare the icing: Beat confectioners' sugar with softened butter and 3 tablespoons of milk. If you want the frosting to be a litter thinner - a glaze or drizzle – add a little more milk as you beat. Remove rolls from oven and let cool slightly before icing. Serve warm.