In a large mixing bowl combine 3 cups of flour, yeast, salt, and sugar. Crack the egg into the flour mixture.
In the microwave safe bowl, warm up the water and butter, until it is mostly melted, about 45 seconds. The water should be comfortable to touch, if it is too hot it will kill the yeast. If you are unsure, use a thermometer and it needs to be less than 139 degrees F for it to not kill the yeast.
Slowly pour the warm water and butter into the flour mixture a little at a time, using a wooden spoon to incorporate it into the flour. Once you have poured all the water in the flour, it will be slightly sticky. When it is difficult to stir, use your hands to knead the dough adding up to ¼ cup of flour if it is still too sticky. Knead the dough for about 5 minutes. With a clean finger you should be able to press into the dough and it does not stick. Shape it into a ball and set it aside for 5-10 minutes. The dough will puff up a little bit.
While it is set aside, mix the brown sugar and cinnamon for the filling. Make sure the butter is spreadable.
Prepare to roll out your dough. On a floured surface place your ball of dough. Use a rolling pin to roll it out to a rectangle that covers a 10x15 mat.
Spread the butter over the rolled out dough and leave a ½ inch border around the edges.
Sprinkle the cinnamon and brown sugar over the butter, pressing it down.
From the long end of the rectangle roll the dough into a long log. It will be about 15 inches long.
Cut the ends of the dough off. This left about 14 inches of the log.
Cut the log into 9 slices. These were about 1.5 inches thick.
In a buttered 9x9 square baking pan, place the rolls in rows of 3.
Cover the rolls and allow them to rise for 1 hour. They will double in size.
Preheat the oven to 350 degrees F. Bake the cinnamon rolls for 20-25 minutes until they are golden on top.
Allow them to cool for 10 minutes. While they are cooling, make the icing. Using a hand mixer, combine the cream cheese, butter, confectioners sugar, and vanilla until it is smooth.
Spread the cream cheese over the warm rolls.