In the bowl of your mixer, mix 2¾ cup flour, yeast, salt, sugar and cinnamon.
In a microwave safe bowl, add milk, butter (cut into small pieces) and water. Microwave for 30 seconds, take the bowl out, stir, then microwave for about 30 more seconds or until the butter is almost melted. Take the bowl out, stir until butter is completely melted. The temperature of this mixture should be 110°F (43°C) or warm to the touch. You can use a thermometer to be more accurate. Add the liquid mixture to the dry ingredients, along with the egg. Start mixing the ingredients together with the dough attachment (or with a spoon if you're not using a mixer), until it comes together. The dough will be quite sticky at this point.
Transfer the dough to a floured surface, start kneading the dough and add sprinkles of flour as necessary, so that it's not too sticky. But don't add too much flour, we want the final dough to be soft and if it's still a tiny bit sticky that's totally fine (I added about two more tablespoons). Knead the dough for 3-5 minutes or until it's soft and when you gently poke it, it bounces back.
Transfer the dough to a lightly greased bowl, cover and place it in a warm place (you can place it in the turned off oven with the light bulb on) until it doubles in size, about 1-1½ hours.
After the dough is doubled in size, transfer it to a lightly floured surface and roll out into a 14x9 inch rectangle.
Spread the softened butter evenly on top.
Sprinkle the top with brown or granulated sugar.
And finally, sprinkle the cinnamon all over. Gently press down with your fingers so that the sugar and cinnamon stick to the butter.
Starting from the 9 inch side, tightly roll the dough up.
Using a knife or a string floss, cut the dough into 1-inch pieces.
Transfer the rolls to a lightly greased (You can use softened butter) baking dish. A pie dish or a 9-inch square baking dish would work perfectly.
Cover the rolls and place them in a warm place until they double in size, about 30-45 minutes.
Preheat the oven to 350°F (177°C) and bake the rolls for 20-25 minutes or lightly browned.
Now we prepare the glaze. In a small bowl, whisk powdered sugar, milk and vanilla until it's smooth. You can add about half a tablespoon more of milk if you would like a runnier consistency. Drizzle some of the icing over the rolls while they're still warm and use the remaining for individual serving.