In a large mixing bowl, combine the warm milk and sugar. Sprinkle the yeast over the milk and let it sit for about 5 minutes until foamy.
In the same bowl, add the melted butter, eggs, and salt. Mix well to combine.
Gradually add the flour, one cup at a time, mixing until a dough forms. If using a stand mixer, continue mixing with a dough hook for about 5-7 minutes until smooth.
Transfer the dough onto a floured surface and knead it for an additional 5 minutes until elastic.
Place the dough in a greased bowl, cover with plastic wrap or a kitchen towel, and let it rise in a warm area for about 1 hour or until doubled in size.
Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
Roll the risen dough out into a large rectangle, about ¼ inch thick.
Spread the softened butter over the dough, then sprinkle the brown sugar and cinnamon mixture evenly on top.
Roll the dough tightly from one end to the other, forming a log. Cut the log into 12 equal pieces.
Place the rolls on the prepared baking sheet, cover them, and let them rise for another 30 minutes.
Bake the rolls in the preheated oven for 25 minutes, or until golden brown.
While the rolls are baking, prepare the icing by whisking together the powdered sugar, milk, and vanilla extract in a small bowl.
Once the rolls are done, let them cool slightly before drizzling the icing on top.