In the bowl of a stand mixer with the dough hook attachment, whisk together the warm milk (about 110°F), yeast, sugar, and 1 cup of flour. Cover the bowl with a towel or plastic wrap and set it aside in a warm place for about 30 minutes.
After 30 minutes, whisk in the eggs, salt and melted butter to the dough mixture. Then, gradually add the remaining 2 1/2 cups of flour until the dough is soft but not runny (depending on the brand of flour, you may need more or less). Attach a dough hook attachment and knead on low speed.
Increase the speed to medium-high and knead the dough until it becomes elastic and pulls away from the sides of the bowl (about 10-15 minutes).
Cover the bowl and let the dough rise for about an hour in a warm place.
For the filling, combine the sugar and cinnamon in a small bowl. Melt the butter in a separate bowl.
Dust a clean work surface with flour and roll out the dough into a rectangle.
Brush the dough with the melted butter, then sprinkle the cinnamon-sugar filling evenly over it. Starting with the long side, roll the dough tightly into a spiral.
Flour a serrated knife and cut the dough into 12 even pieces. Place the rolls onto a buttered 14 inch round baking pan, then cover and let them rest for 1 hour.
Meanwhile, preheat the oven to 350°F. Once the rolls have doubled in size, bake them for 35-40 minutes or until golden brown.
For the icing, beat together the softened cream cheese, powdered sugar, and vanilla extract with an electric hand mixer until smooth and creamy.
Once the cinnamon rolls are out of the oven, spread the icing over them and serve warm. Enjoy!