To Make dough: 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 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 oven.
To Make Filling: 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 cut into 9 (large) even piece. Place in a lightly greased 9-inch or square pan (or cut into 12 small pieces and place in a 9x13 pan) and lightly cover with aluminum foil or plastic wrap.
Place the cinnamon buns in the oven to rise for 20 minutes (with the oven off). After 20 minutes, Keep the buns in the oven (REMOVE THE FOIL OR PLASTIC) and turn on the oven to 375F. Bake the cinnamon rolls for an additional 15-20 minutes or until golden. Remove from oven and top with glaze.
To make glaze: Mix the powdered sugar, vanilla, and 2 Tablespoons milk together until smooth and lump free. Drizzle over warm rolls.