In a large mixing bowl, combine 2 cups of flour and yeast. Set aside.
In a small saucepan, heat milk, sugar, butter, and salt until warm and butter is almost melted. Add to flour mixture along with the egg. Beat with an electric mixer on low speed for 30 seconds, then on high speed for 3 minutes. Stir in remaining flour.
Turn dough onto a floured surface and knead until smooth and elastic. Shape into a ball and place in a greased bowl, turning once to grease the surface. Cover and let rise in a warm place until doubled in size, about 1 hour.
Punch dough down. Turn onto a lightly floured surface and roll into a 12x8 inch rectangle. Spread with softened butter. Combine brown sugar and cinnamon, then sprinkle over buttered dough.
Roll up dough starting from a long side. Pinch seam to seal. Cut into 12 slices and place in a greased baking dish. Cover and let rise until nearly doubled, about 30 minutes.
Bake in a preheated 375°F oven for 20-25 minutes or until golden brown. Cool slightly.
For icing, combine powdered sugar, milk, and vanilla extract. Drizzle over warm rolls. Sprinkle with rainbow sprinkles.