1. In a large mixing bowl, combine 2 cups of flour and yeast.
2. In a saucepan, heat milk, sugar, butter, and salt until warm (120-130°F).
3. 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.
4. Turn dough out onto a lightly floured surface. Knead in enough remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape into a ball. Place in a greased bowl; turn once. Cover; let rise in a warm place until double in size (about 1 hour).
5. Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest for 10 minutes. Meanwhile, lightly grease two 9x1-½-inch round baking pans or a 13x9x2-inch baking pan.
6. 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, cut sides down, in prepared pans. Cover; let rise until nearly double (about 30 minutes).
7. Bake in a 375°F oven for 20 to 25 minutes or until golden brown. Cool slightly.
8. For icing, in a small mixing bowl, combine powdered sugar, butter, and vanilla. Add enough milk to make icing of drizzling consistency. Drizzle over warm rolls.