1. In a large bowl, combine flour, sugar, yeast, and salt. Mix well and set aside.
2. In a small saucepan, heat the milk and butter over medium-low heat until the butter melts. Remove from heat and let it cool to lukewarm.
3. Add the milk mixture to the dry ingredients along with the egg and vanilla extract. Mix until a dough forms.
4. Place the dough on a floured surface and knead for about 5-7 minutes until smooth and elastic.
5. Transfer the dough to a lightly greased bowl, cover with plastic wrap, and let rise in a warm place for about 1-1 ½ hours, or until doubled in size.
6. For the filling, combine brown sugar and cinnamon in a small bowl.
7. Once the dough has risen, punch it down and roll it out on a floured surface into a large rectangle, about 16×10 inches.
8. Spread the softened butter evenly over the dough. Sprinkle the cinnamon-sugar mixture over the butter, covering the entire surface.
9. Starting from the long side, roll the dough tightly into a log.
10. Cut the log into 12 equal pieces and place them in a greased 9×13-inch baking dish.
11. Cover the dish with plastic wrap and let the rolls rise again for about 30 minutes.
12. Preheat the oven to 350°F (175°C).
13. Bake the rolls for 20-25 minutes, or until golden brown.
14. While the rolls are baking, prepare the glaze by mixing the powdered sugar, milk, and vanilla extract until smooth.
15. Once the rolls are out of the oven, let them cool slightly and drizzle the glaze over the top.