Make the dough: Add dough ingredients to bread machine in order listed. Select dough cycle and start. May need to let dough rise for 10-20 more minutes after cycle is complete.
Grease the bottom and sides of a 9×13 inch baking dish or line with parchment paper. Turn the risen dough out onto a lightly floured work surface, punch it down and, using a rolling pin, roll into a 13×18 inch rectangle. Make sure the dough is smooth and evenly thick.
For the filling: Spread the softened butter all over the dough. In a small bowl, toss the cinnamon and sugar together until combined and then sprinkle evenly over the dough. Tightly roll up the dough to form an 18-inch-long log. If some filling spills out, just sprinkle it on top of the rolls. Cut into 12 even rolls--dental floss works really well. Arrange them in the prepared baking pan, cut sides up. Cover the rolls very tightly with aluminum foil (no rolls exposed- this dries out your dough!) and stick in the refrigerator** for 8-12 hours.
Remove rolls from the refrigerator and let rise in a warm place until they are puffy, about 1-2 hours.
Make the icing: In a medium bowl using a handheld or stand mixer fitted with a paddle attachment, beat the cream cheese on medium speed until smooth and creamy. Add the confectioners' sugar, milk, and vanilla. Beat on low speed for 30 seconds, then switch to high speed and beat for 2 minutes.
After the rolls have risen, preheat the oven to 375°F (191°C). Bake for 12-15 minutes until the tops are lightly brown, then cover loosely with aluminum foil and bake for another 12-15 minutes until internal temp reads 190-200°F. Remove pan from the oven and frost right away with about 1/4 to 1/3 of the icing using a knife. Allow to cool for 10-15 minutes before completely icing rolls and serve immediately.
Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 3 days. Storing in the refrigerator can dry out the rolls.