Make the Dough:
In a large bowl, whisk 3 cups of flour, sugar, yeast, and salt.
In a microwave-safe bowl or saucepan, heat the water, milk, and butter until the butter is melted and the mixture is warm but not hot.
Whisk the egg into the warm liquid.
Pour wet mixture into dry and stir with a wooden spoon until a thick dough forms.
Add another 1 cup of flour and continue mixing until it begins to form a ball.
Knead on a floured surface (or in the bowl) for about 8–10 minutes, adding more flour as needed, until the dough is soft, elastic, and slightly tacky.
Cover and let rest for 20 minutes.

Shape the Rolls:
Roll the dough into a 14×10-inch rectangle on a lightly floured surface.
Spread the softened butter over the dough, leaving a 1/2-inch border.
Sprinkle brown sugar evenly over butter, then sprinkle cinnamon on top.
Roll up the dough tightly from the long side into a log.
Trim ends and cut into 12 even rolls using a sharp knife or dental floss.
Place rolls cut-side up in a greased 9×13-inch baking dish, with slight space between them.
Cover and let rise in a warm place for 45 minutes, or until puffy and doubled in size.

Bake:
Preheat the oven to 350°F (175°C).
Uncover the rolls and bake for 25–30 minutes, until golden brown. Remove from oven and set aside.

Make the Icing:
In a medium bowl, cream together cream cheese, butter, powdered sugar, salt, and vanilla until smooth and fluffy.

Finish:
Spread icing over warm rolls so it melts into the crevices. Serve warm and enjoy!