In a large bowl, mix the flour, the sugar, salt, and yeast together until evenly dispersed. Set aside. 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 with hand or with a stand mixer for 3-4 minutes or until the dough is no longer sticky.
Place in a lightly greased bowl and let rest for about 5 minutes.
Preheat the oven to 200 degrees and turn off after 10 minutes or just before placing rolls in the oven.
Allow the dough to rest for 5 minutes, and then roll it out into a 15x9" rectangle.
Spread the softened butter over the dough, and sprinkle with the cinnamon and sugar.
Roll the dough into a log starting at the wide end and pinch the seam to seal it.
Slice 9 even pieces out of the log and place in a greased 8x8 (or 9x9) square baking dish.
Make sure the oven is off and place the cinnamon buns in the oven to rise for 20 minutes.
Keep the buns in the oven and turn on the oven to 375F.
Bake for 15-20 minutes or until golden.