In a large mixing bowl, combine 2 cups of flour and yeast.
In a saucepan, heat milk, butter, sugar, and salt until warm and butter is almost melted.
Add milk mixture to flour mixture; add eggs. Beat with an electric mixer on low speed for 30 seconds, scraping bowl constantly. Beat on high speed for 3 minutes. Stir in as much of the remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape dough into a ball. Place in a greased bowl; turn once. Cover; let rise in a warm place until double in size (about 1 hour).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest for 10 minutes. Meanwhile, lightly grease two baking pans or a large baking dish.
Roll each half of dough into a 12x8-inch rectangle. Spread with softened butter. Combine brown sugar and cinnamon; sprinkle over butter. Roll up each rectangle, starting from a long side. Seal seams. Slice each roll into 12 pieces. Place slices in prepared pans. Cover and let rise until nearly double in size (about 30 minutes).
Preheat oven to 375°F (190°C). Bake rolls for 20 to 25 minutes or until golden brown. Cool slightly.
For icing, in a small bowl combine powdered sugar, milk, and vanilla. Drizzle over warm rolls.