In a large mixing bowl, combine 2 cups of flour and yeast.
In a saucepan, heat milk, sugar, butter, and salt until warm (120-130°F).
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 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).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover; let rest for 10 minutes. Roll each half into a 12x8-inch rectangle.
Spread with softened butter. Combine brown sugar and cinnamon; sprinkle over buttered dough. Roll up each rectangle, starting from a long side. Seal seams. Slice each roll into 12 pieces. Place slices, cut sides down, in greased baking pans. Cover; let rise until nearly double in size (about 30 minutes).
Bake in a 375°F oven for 20 to 25 minutes or until golden brown. Cool slightly.
For icing, in a small mixing bowl beat butter, powdered sugar, and vanilla with enough milk to make icing of drizzling consistency. Drizzle over warm rolls.