1. To prepare rolls, combine milk, 3 tablespoons melted butter, 1 tablespoon granulated sugar, and yeast in a large bowl; let stand 5 minutes. Weigh or lightly spoon flour into dry measuring cups. Add egg and remaining granulated sugar to bowl. Stir in 4.5 ounces (1 cup) flour; let stand 10 minutes.
2. Add 11.25 ounces (about 2 1/2 cups) flour and salt to milk mixture; stir until a soft dough forms (dough will be sticky). Turn out onto a lightly floured surface. Knead until smooth and elastic (about 6 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands. Place dough in a large bowl coated with cooking spray; turn to coat top. Cover and let rise in a warm place (85°), free from drafts, 35 minutes or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rise 35 minutes or until doubled in size. Punch dough down; cover and let rest 5 minutes.
3. Combine brown sugar and cinnamon. Turn dough out onto a lightly floured surface; roll dough into an 18 x 11–inch rectangle. Brush remaining 3 tablespoons melted butter over dough; sprinkle evenly with brown sugar mixture. Beginning at one long side, roll up dough tightly, jelly-roll fashion; pinch seam to seal (do not seal ends of roll). Cut dough into 18 (1-inch) slices. Arrange 9 slices, cut sides up, in each of 2 (8-inch) square baking dishes coated with cooking spray. Cover and let rise 35 minutes or until doubled in size.
4. Preheat oven to 350°.
5. Uncover rolls. Bake at 350° for 22 minutes or until lightly browned. Cool 10 minutes in dishes on a wire rack. Turn rolls out onto wire rack; cool 5 minutes. Turn rolls over.
6. To prepare icing, combine 3 tablespoons softened butter and cream; stir with a whisk. Stir in vanilla. Gradually add powdered sugar; stir until blended. Spread icing over rolls; serve warm.