To Make the Dough:
Spray a baking dish with non-stick spray. Set aside.
Place the warm water in a large bowl. Sprinkle the yeast on top and whisk to combine. Let the mixture rest and activate for 10 minutes.
Once the yeast is active and bubbly, add the milk, sugar, salt, vanilla extract, and egg. Whisk to combine.
Gradually stir in the flour. If using a stand mixer, use the dough hook. If mixing by hand, continue with a wooden spoon. Add 1 cup of flour and allow the dough to come together, then add the softened butter. Continue mixing until the butter is mostly incorporated. Add the remaining flour (up to 1 and ¼ cups) and mix until the dough has mostly come together. The dough should pull away from the sides of the bowl easily and be slightly elastic.
Transfer the dough to a floured surface and knead for 8-10 minutes, adding flour 1 Tablespoon at a time if the dough is too sticky, until it is smooth and elastic. The dough should resemble the texture of PlayDoh. Allow the dough to rest for 10 minutes while preparing the filling.
To Make the Filling:
In a medium-sized bowl, combine the granulated sugar, brown sugar, and cinnamon. Whisk or stir until well blended. Set aside.
Roll the dough into a 9″x12″ rectangle. The dough may be elastic, so be patient and continue rolling until you reach the desired size.
Brush the melted butter evenly over the dough, leaving a ½” border around the edges. Sprinkle the cinnamon-sugar mixture evenly over the melted butter.
Assemble and Bake the Rolls:
Starting from one long side, roll the dough into a tight 12″ log. Use your fingers or palms to gently squeeze the log to even out the ends. Using a sharp serrated knife, cut the log into 6 even slices, ensuring the seam side is down.
Place the slices cut-side down in the prepared pan. Gather any spilled cinnamon-sugar and sprinkle it over the rolls. Press down gently to flatten slightly. Cover loosely with foil or a kitchen towel and allow the rolls to rise in a warm, draft-free place for 1 and ½ hours.
Preheat the oven to 350°F (177°C). Once the rolls have risen, bake them for 28-32 minutes or until they just start to brown on top. Cover with foil to prevent excessive browning if desired. Remove from the oven and let them cool for 15 minutes while you prepare the cream cheese frosting.
To Make the Cream Cheese Frosting:
In a large bowl with a handheld mixer or a stand mixer fitted with a paddle attachment, beat the cream cheese and butter on medium-high speed until smooth.
Turn the mixer speed to low and gradually add the powdered sugar. Add the vanilla extract and a pinch of salt. Continue mixing until smooth.
Spread the cream cheese frosting over the slightly cooled rolls. Serve immediately.
Storage: Leftover cinnamon rolls stay fresh in an airtight container at room temperature for up to 3 days or in the refrigerator for up to 5 days. Reheat in the microwave for 8-10 seconds before serving.