Add 2 cups flour, sugar, dry yeast, and salt in the bowl of a stand mixer fitted with the dough hook. Whisk to combine.
Place water, milk and butter in a microwave-safe measuring cup. Heat, in 30 second increments, until the water reaches 120°-130°F (think hot bath water, use an instant read thermometer for best results). This will take approximately 1 1/2 minutes, depending on your microwave. Butter won’t totally melt.
Add water mixture to dry ingredients with the egg. Mix on medium speed for 2 minutes, scraping the sides of the bowl a few times.
Add 1 cup more flour, beat an additional 2 minutes at high speed, scraping bowl occasionally. Mix in just enough remaining flour so the dough forms into a ball. The dough will be slightly sticky but not wet.
Turn the dough out onto a lightly floured surface. Knead until smooth and elastic, about 6-8 minutes. You know it’s done when it forms a cohesive ball and springs back when lightly pressed with 2 fingers.
Cover and let rest for 10 minutes.
Meanwhile, make the filling by stirring together the sugar and cinnamon and melting the butter.
Roll dough into an approximately 11×17-inch rectangle using a rolling pin. Brush the butter all over the dough in a thin layer. Sprinkle with cinnamon sugar and then sprinkle with shredded zucchini.
Beginning at the long side of the dough, roll up tightly. Slice into 12 equal pieces.
Place rolls in a 9×13-inch baking pan that’s been sprayed with nonstick cooking spray.
TO BAKE IMMEDIATELY:
Cover rolls and let double in size, about 1 hour, then skip to baking directions below.
TO MAKE OVERNIGHT CINNAMON ROLLS:
Cover with plastic wrap that has been sprayed with nonstick cooking spray. Place the rolls in the refrigerator overnight.
In the morning, preheat oven to 200°F, place the rolls in the oven then immediately turn the oven off. Let the rolls sit in the warm oven (no peeking) for one hour to rise.
After the hour, remove the rolls from the oven.
TO BAKE:
Preheat oven to 375°F. Bake cinnamon rolls for about 25-35 minutes or until golden brown and the center is cooked through. Cool 20 minutes before frosting.
MAKE THE FROSTING:
Beat butter and cream cheese in a large bowl until creamy using a hand mixer. Mix in powdered sugar and vanilla.