In a bowl, sprinkle yeast over warm water and add sugar. Let stand 10 minutes, until foamy. Heat milk and 6 tablespoons butter over medium heat until warm, 105F to 115F (butter may not melt completely).
Mix dissolved yeast, milk mixture, 1/2 cup sugar, and eggs in large bowl. Beat with a wooden spoon until well combined. Add 3 cups flour and salt and beat until smooth. Stir in 2 cups flour, 1 cup at a time, to make a soft dough. On lightly floured surface, knead dough 3 to 5 minutes until smooth and elastic, adding more flour if needed. Place dough in a large greased bowl, turning to grease top. Cover and let rise in a warm, draft-free place until doubled in bulk, 1 1/2 hours.
Melt remaining 1/4 cup butter in saucepan. Combine 1/2 cup sugar and cinnamon and set aside.
Grease a 15 x 10 inch jelly-roll pan. Punch down dough. On lightly floured surface with floured rolling pin, roll dough into a 20 x 14 inch rectangle. Brush with 2 tablespoons melted butter and sprinkle with cinnamon sugar. From one long side, roll up jelly-roll style; pinch seam to seal. Cut with a serrated knife into 18 slices, about 1 inch thick. Arrange slices cut side down in prepared pan. Brush remaining melted butter evenly over tops of rolls. Cover with wax paper and let rise until doubled, 45 to 60 minutes.
Heat oven to 350F (180 C). Uncover rolls and bake 22 to 25 minutes, until lightly browned. Cool 15 minutes on wire rack. Remove from pan.
Meanwhile, for the icing, beat powdered sugar and butter in mixer bowl at medium speed until blended. Beat in 3 tablespoons milk, vanilla, and pinch cinnamon until smooth. With a spoon, drizzle icing evenly over tops of rolls.