Warm the milk until barely warm, about 100–105°F.
Stir in the sugar, then sprinkle yeast over milk. Let it foam 5–10 minutes.
Whisk in melted butter, eggs, and salt until smooth.
Add 3 cups flour. Mix to a shaggy dough.
Add remaining flour gradually. Knead until smooth and slightly tacky, 6–8 minutes.
Place in a greased bowl, cover, and let rise until doubled, 60–75 minutes.
Mix brown sugar and cinnamon for the filling. Set aside.
Punch down dough. Roll into a 12x18-inch rectangle on a floured surface.
Spread softened butter over dough, leaving a small border.
Sprinkle cinnamon sugar evenly. Press lightly to adhere.
Roll up tightly from the long side. Pinch the seam to seal.
Slice into 12 rolls using a sharp knife or floss.
Arrange in a greased 9x13-inch pan, spaced slightly apart.
Cover and proof until puffy, 25–35 minutes.
Bake at 350°F until golden and set, 22–26 minutes.
Beat cream cheese, powdered sugar, vanilla, and milk until creamy.
Spread icing over warm rolls. Serve.