In a large bowl, mix flour, yeast, sugar, and salt. Stir in warm milk, water, melted butter, and egg. Mix until dough forms.
Knead the dough on a lightly floured surface for about 10 minutes until smooth. Let it rest in a greased bowl, covered, for an hour or until doubled in size.
Roll the dough into a rectangle. Spread softened butter across the surface. Mix brown sugar and cinnamon, then sprinkle over the butter.
Roll it up tightly, slice into 12 rolls, and place them in a greased baking dish. Cover and let them rise for 30 minutes.
Preheat your oven to 350°F (175°C). Bake rolls for 20–25 minutes until golden.
Beat cream cheese, butter, powdered sugar, and vanilla until smooth. Spread frosting over warm rolls and enjoy.