In a mixing bowl, combine warm milk and active dry yeast. Let it sit for about 5 minutes until bubbly.
In a large bowl, whisk together flour, sugar, and salt. Make a well in the center, and add in the yeast mixture, melted butter, and eggs. Mix until a dough forms.
Knead the dough on a lightly floured surface for about 6-8 minutes until smooth and elastic.
Place the dough in a greased bowl, cover with plastic wrap, 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).
After the dough has risen, roll it out on a floured surface into a rectangle (about 16x12 inches).
Spread the softened butter for the filling over the dough, then sprinkle the brown sugar and cinnamon evenly over the top.
Starting from the long edge, roll the dough tightly into a log. Cut the log into 12 equal pieces.
Place the rolls on a greased baking sheet, ensuring they are spaced apart. Cover and let them rise for an additional 30 minutes.
Bake in the preheated oven for 20-25 minutes, or until golden brown.
While the rolls are baking, prepare the icing by whisking together powdered sugar, milk, and vanilla extract until smooth.
Once the rolls are done, let them cool slightly before drizzling the icing over the top.