Into a stand mixer, mix together the milk, instant dry yeast, sugar, eggs, butter, and salt. Switch to the dough hook in the stand mixer or with a large wooden spoon in a bowl, add flour, and mix until a dough forms. Cover the bowl with a towel and rest for 1 hour.
Mix together brown sugar and cinnamon until combined. Transfer the dough onto a floured work surface and roll into a rectangle. Brush the melted butter over the dough. Sprinkle the cinnamon sugar mixture evenly.
Roll the dough into an 18-inch log. Using a sharp knife, evenly cut the log into 12 rolls. Transfer the rolls to the baking dish. Cover the rolls with a towel and let them rise for 30 minutes.
Bake the rolls at 350° for 25-28 minutes or until golden brown on top. Let them cool for 15 minutes before icing. Whisk together the powdered sugar, 2 tbsp of milk, butter, and vanilla until smooth. Pour the icing on top.