To make dough:
In the bowl of mixer, dissolve yeast in warm milk & let stand until foamy. (10 min.)
Add granulated sugar, eggs, flour & salt.
Use dough hook & knead on low speed, adding a little flour if needed until ingredients come together.
Toss in the butter & continue to knead until the dough is smooth & springy. (7 min)
Form dough into a ball & put in an oiled bowl & cover with plastic wrap.
Let dough rise at room temperature until it doubles. (2 hours)
Butter 9x13" baking pan.
Punch down dough & turn onto a lightly floured surface.
Cut dough in half.
Roll out into a rectangle- about 10x7"
Spread with half the soft butter.
Sprinkle with cinnamon/sugar mixture, then cover with brown sugar & then sprinkle with cinnamon.
Starting on the long side- roll dough into a log. Pinch ends to close.
Slice into 8 equal rounds. Arrange slicked side down in pan.
Repeat with remaining dough & ingredients.
Cover pan loosely with plastic wrap & let stand in warm, draft-free spot until puffy. (at least an hour but I refrigerated mine overnight & then let stand at room temp for 1 hour before baking)
Position rack in middle of oven. Preheat to 400 degrees.
Brush buns with melted butter. Sprinkle with additional brown sugar.
Bake until golden brown 20-25 minutes.
Let cool slightly in pan on a wire rack.
Spread with frosting while still warm.
For the frosting:
Combine all ingredients in small bowl.
Whisk until combined. Keep adding small amounts of confectioners sugar or milk until you reach the desired consistency that you are after. I wanted mine slightly thin so I could drizzle. You may want yours to spread on so use more confectioners sugar for that.