For the rolls: In a microwave safe bowl, combine 3/4 cup milk and 4 tablespoons butter. Heat on high for 1 1/2 minutes. (Butter should be partly melted.) Stir in remaining 1/4 cup cold milk and set aside.
In a large mixing bowl (or stand mixer with paddle attachment) whisk together 2 cups flour, sugar, yeast, and salt. Add milk mixture and egg to the flour mixture while the beater (or paddle) is running. Beat for about 1 minute until ingredients are completely incorporated. Switch to your dough hook attachment and add remaining flour only until dough barely pulls away from the sides of the bowl. (It should still be very soft and slightly sticky - do not add too much flour!.) Knead for 5 minutes (if using stand mixer, continue to use dough hook, if using a hand mixer transfer dough to a floured surface and knead by hand for 5 minutes).
Transfer dough out onto a floured surface, roll one or twice to lightly coat in flour, and allow to rest for about 10 minutes. Roll dough out into a large rectangle (about 12x14 inches). Use the back of a spoon or a spatula to spread cream cheese all over the top surface of the dough. Fold dough in thirds (like a letter) and then in thirds again. Turn dough once or twice to lightly coat in flour again and smooth out the folds. Allow to rest for 5-10 minutes.
Roll dough out into a rectangle again (about 12x14 inches). Spread softened butter over the dough. Whisk together cinnamon and sugar. Sprinkle buttered dough with cinnamon-sugar mixture. Roll tightly starting and ending with the longer ends. Cut dough log into 12 equal sized pieces and place side-by-side in a greased 9x13 inch pan. Cover pan with plastic wrap or drape a towel over the top and place in warm place for 20-30 minutes. Bake at 350 degrees for 15-20 minutes until golden brown around the edges and on top.
For the frosting: Melt butter in a medium sauce pan over medium heat. Once butter is melted, continue to whisk over medium heat until butter turns from yellow to a golden color. Remove from heat.
In a medium bowl cream together butter, cream cheese for 2-3 minutes until very fluffy and creamy.
Mix in powdered sugar, one cup at a time, adding 1-2 tablespoons of milk after each addition of powdered sugar. Once all of the powdered sugar is incorporated, mix in vanilla and more milk (1 tablespoon at a time) until icing is a smooth spreadable consistency.
Spread immediately on cinnamon rolls, or store chilled in airtight container up to five days.
