Preheat oven to 200. In the bowl of a stand mixer, combine the flour, the sugar, salt, and yeast and mix.
In a small microwavable bowl, heat the water, milk, and butter together in the microwave until the butter is melted (about 30-45 seconds). Stir the butter mixture into the flour mixture. Add the egg and knead by hand or with the stand mixer for 3-4 minutes or until the dough is no longer sticky. Add up to 1/4 c additional flour if needed so dough does not stick to your finger. Gather the dough into a ball and place back into the bowl to rest for 5 minutes.
After the dough has rested for 5 minutes, roll it out in a 15x9 inch rectangle. Spread the softened butter on top. Mix together the cinnamon and sugar and sprinkle it all over the dough. Roll up the dough tightly and pinch the seam together. Use floss or a very sharp knife to cut into 10-12 pieces. Place in a greased 9x13 pan and cover lightly with foil.
TURN THE OVEN OFF and place the cinnamon rolls in the oven to rise for 20 minutes. Prepare the frosting while rolls rise.
After 20 minutes, remove the foil (leave the pan in the oven!) and turn on the oven to 375F. Bake the cinnamon rolls for 15-20 minutes or until golden. Remove from oven and top with frosting while warm.