Heat milk and water to between 110F to 120F. It's easy to do this in the microwave. Heat for 30 seconds, check the liquid with a digital thermometer and heat another 20 to 30 seconds if needed. Note: don't put liquids hotter than 120 degrees in the machine. It will ruin the proof.
Load ingredients according to the instructions that came with your bread machine.
Set the machine to the dough cycle and let it run.
When the alarm sounds, check the dough for elasticity and add water or flour as needed.
Lightly oil your work surface.
When the bread machine finishes, remove the dough to your work surface and shape into a ball. Cover completely with a cotton cloth or plastic wrap and let it rest for 10 to 20 minutes.
While the dough rests, mix the softened butter, brown sugar and cinnamon together for spreading.
Roll the dough out into a rectangle shape about 12 by 24 inches. You want it to be about ¼ inch thick.
Spread the filling evenly onto the dough, leaving a ¼ inch border.
Roll up the dough using gentle pressure to keep the roll even. Your roll should be nice and tight. Pinch the seam together to seal it securely
Measure and mark the dough into 12 equal portions. Cut the portions apart with string or a pastry scraper blade.
Line a 9 x13 baking dish with parchment paper and place cinnamon rolls inside in rows of three.
Cover with a cotton towel or cloth (not terrycloth) and allow to proof until doubled in size (approximately 30 minutes)
After proofing, place cinnamon rolls on the center rack of a 375F preheated oven and bake 20 to 25 minutes until golden brown.
After baking, place the rolls on a wire rack to cool. While the cinnamon rolls cool, whip the icing ingredients together until smooth. Spread icing onto the cooled rolls with a silicone spatula and serve.