Add the lukewarm milk, sugar, cooled butter, eggs, and yeast in the bowl of a stand mixer. Mix everything together using the paddle attachment.
Add 4½ cups of flour and salt to the mixture. Mix on low just until a ball of dough has started to form. Use your hands to remove the dough from the paddle attachment. Remove the paddle and replace it with the dough hook. Mix (knead) on medium for 6 to 7 minutes until the dough is soft and does not stick to the bowl, adding the remaining ¼ cup of flour as needed.
Remove the dough from the bowl and smooth it into a ball with your hands. Spray a large bowl with cooking spray or softened butter. Place the dough in the bowl and turn it over a couple of times to coat it with oil. Cover with plastic wrap and place in a non-drafty, slightly warm area until doubled in size, usually about 2 hours. Or, place in the fridge overnight for at least 12 hours.
Remove the plastic wrap and dump the dough onto a cool surface that is lightly floured. Use a rolling pin to roll the dough into an approximate 10"x15" rectangle.
Meanwhile, place the softened butter (7 tbsp), the two sugars, and cinnamon in a medium-sized bowl. Use a hand mixer to mix until all ingredients are well incorporated. If not using immediately, place in the refrigerator until ready to use.
Use a butter knife to smear the butter/sugar/cinnamon mixture all over the dough, leaving about ½-inch around the edges.
Beginning on the long side of the rectangle, roll up the dough completely, and gently pinch the edges together to seal. Use a large, sharp knife to cut 12 rolls from the log.
Grease a 9"x13" baking dish (preferably metal) with cooking spray or softened butter. Place the rolls in the pan. Loosely cover with plastic wrap, and place in a non-drafty, slightly warm area until the rolls have nearly doubled in size, usually 60 to 90 minutes.
Preheat oven to 350°F.
Remove the plastic wrap and bake on the center rack for 20 minutes, or until the rolls are lightly browned on top and fully cooked. Remove from oven and let cool for about 10 minutes.
Use a large spoon to pour the icing over the warm rolls.