Stir together 1 3/4 cups flour, yeast, and salt in a large mixing bowl.
Heat milk, water, butter, and sugar until very warm ( 120 deg~130 deg F).
Add liquid ingredients to flour mixture; beat until smooth about two minutes on electric mixer.
Add egg and 1/2 cup more flour and beat another 2 minutes.
Gradually add enough more flour to make a soft dough.
Turn out onto floured board; continue to work in flour until dough can be kneaded. Knead until smooth and elastic, but still soft ( about 5 minutes).
Cover; let rest about 20 minutes.
Roll dough into a 13x9-inch rectangle. Spread with 2 tablespoons softened butter or margarine. Sprinkle with mixture of 1/2 cup (or more)firmly packed brown sugar and 2 teaspoons(or more) cinnamon. Beginning with long side, roll dough up tightly jelly-roll fashion. Cut roll into 12(1inch) slices. Place slices in a greased 13x9x2- inch baking pan or greased muffin cups. Bake at 375 deg F, 15~ 20 minutes.
For cream cheese frosting. Cream butter and cream cheese. Add half of powdered sugar, mix. Add vanilla, milk and remaining powdered sugar. Mix until well combined. Frost WARM rolls.
*Coat baking pan with softened butter and sprinkle with brown sugar.