In a saucepan, heat the milk, sugar, and 1/2 cup of butter over medium heat until the butter is melted and the mixture is warm (not boiling). Remove from heat.
Sprinkle yeast over the warm milk mixture and let it sit for 5-10 minutes, until foamy.
In a large mixing bowl, whisk together the eggs and then add the yeast mixture.
Gradually mix in the flour and salt using a wooden spoon until a dough forms.
Knead the dough on a floured surface for about 5 minutes, until smooth. Place dough in a greased bowl, cover with plastic wrap, and let rise in a warm place for 1 hour, or until doubled in size.
After rising, punch down the dough, and roll it out into a rectangular shape (about 16x24 inches).
Spread the softened butter over the dough. In a bowl, mix brown sugar and cinnamon, then sprinkle evenly over the butter.
Starting from one long end, roll tightly to form a log. Pinch the seam to seal.
Cut the log into 12 equal pieces and place in a greased 9x13 inch baking pan. Cover and let rise for another 30 minutes.
Preheat the oven to 350°F (175°C). Bake the rolls for 20-25 minutes, until golden brown.
While the rolls are baking, prepare the icing. In a bowl, whisk powdered sugar, vanilla extract, and enough milk to reach desired consistency.
Once the rolls are out of the oven, drizzle the icing over the rolls while they are still warm.