Whisk the flour, sugar, salt, and yeast together in a large bowl. Set aside.
Combine the milk, water, and butter together. Microwave or use the stove and heat until the butter is melted and the mixture is warm to touch. Pour into the dry ingredients, add the egg, and stir until it forms a soft dough.
On a lightly floured surface using floured hands, knead the dough for 3 minutes. Place in a lightly greased bowl, cover loosely, and let rest for about 10 minutes.
After 10 minutes, roll the dough out in a 14×8 inch rectangle. Spread the softened butter on top. Mix together the cinnamon and sugar. Sprinkle it all over the dough. Roll up the dough tightly. Cut into 10-12 even rolls and arrange in a lightly greased 9-inch round cake pan or pie dish.
Tightly cover the rolls with aluminum foil or plastic wrap and allow to rise in a warm, draft-free environment for 60-90 minutes.
After the rolls have doubled in size, preheat the oven to 375°F (190°C). Bake for 25-28 minutes until lightly browned. If you notice the tops are getting too brown too quickly at about 15 minutes, loosely cover the rolls with aluminum foil and continue baking. (do not worry browning is normal and taking them out to put on foil won't hurt them)
Whisk the confectioners’ sugar, vanilla extract, and milk together. Drizzle or spread over warm rolls.Cover leftover frosted or unfrosted rolls tightly and store for up to 3 days at room temperature or up to 5 days in the refrigerator