Measure out 1/2 cup of flour and then set it aside.
In the bowl of your stand mixer add 2 1/4 cups of flour, sugar, salt, and yeast together until evenly mixed with a rubber spatula and then set on the mixer stand.
In a double boiler (if you're not sure how to make one, learn how to make one at home), combine the water, milk, and butter until the butter is melted and the mixture is hot to the touch. You can also do this step in the microwave if preferred.
Add the dough hook to the mixer.
Pour the butter mixture into the flour mixture and mix on low speed until combined.
Next, add the egg and the leftover flour to the cinnamon roll dough mixture to make a soft dough. Start with 1/3 cup and work up to 1/2 cup until the dough is ready. Keep the mixer on low speed while adding the flour and then up to medium speed to finish making the dough.
The dough is ready when it pulls away from the side of the bowl and has an elastic consistency.
Remove the dough from the stand mixer and Knead the dough for about 3-4 minutes on a floured surface.
Grease the stand mixer bowl, add the dough, and cover with a kitchen towel. Let the dough rise and rest for about 10 minutes.
While the dough is rising, make the cinnamon roll filling. In a bowl, mix the butter, cinnamon, and brown sugar. Make sure the butter is soft before adding it to the bowl.
Once your dough is ready, roll it out in a rectangle. Spread the filling mixture on top.
Roll up the dough tightly by squeezing the edges closed, then cut into 1- 1 1/2 inch wide pieces.
Spray a circle baking pan with nonstick cooking spray.
Place cinnamon rolls in the pan. You want the top to be where you can see the filling on the inside.
Place the baking pan covered in a dish towel in a nondrafty area, and let the dough rise until the cinnamon rolls have about doubled in size. This will take 60-90 minutes.
Once they have doubled in size, preheat the oven to 375 degrees. When the oven is ready, place the cookies in the oven and bake for 25-30 minutes until they are light golden brown.
In a bowl, combine the vanilla extract, cream cheese, and butter and cream them together. Once creamed, add powdered sugar and mix well. You can also use a stand mixer for this.
Once the rolls have been removed from the oven, frost using half the amount made. Let them cool, and then frost again.