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Whisk the warm milk, 2 Tablespoons sugar, and the yeast together in the bowl of your stand mixer fitted with a dough hook or paddle attachment. Cover and allow mixture to sit for about 5 minutes or until foamy on top. *If you do not own a mixer, you can do this in a large mixing bowl and in the next step, mix the dough together with a large wooden spoon/rubber spatula. It will take a bit of arm muscle. A hand mixer works, but the sticky dough repeatedly gets stuck in the beaters. Mixing by hand with a wooden spoon or rubber spatula is a better choice.*
On medium speed, beat in the remaining sugar (which should be about 1/2 cup) and the softened butter until it is slightly broken up. Add the eggs and salt and beat on medium speed until combined. The butter won’t really be mixing into the mixture, so don’t be alarmed if it stays in pieces. Switch the mixer down to low speed and with it running, add 1 cup of flour at a time, making sure it’s fully incorporated before adding the next. After 4 cups have been added, add the last 1/2 cup and beat until the dough comes together and pulls away from the sides of the bowl, about 3 minutes. Dough will be soft.
Keep the dough in the mixer and beat on low speed for an additional 3 minutes or knead by hand on a lightly floured surface for 3 minutes.
Lightly grease a large bowl with oil or use nonstick spray. Place the dough in the bowl, turning it to coat all sides in the oil. Cover the bowl with aluminum foil, plastic wrap, or a clean kitchen towel. Allow the dough to rise in a relatively warm environment for 2 hours or until double in size. (I always let it rise on the counter and it takes about 2 hours. For a tiny reduction in rise time, see my answer to Where Should Dough Rise? in my Baking with Yeast Guide.)
Grease the bottom and sides of a metal or glass 9×13 inch baking dish or line with parchment paper.
Punch down the dough to release the air. Place dough on a lightly floured work surface and using a lightly floured rolling pin, roll dough into a 12×18 inch rectangle. Make sure the dough is smooth and evenly thick. If the dough keeps shrinking as you roll it out, stop what you’re doing, cover it lightly, and let it rest for 10 minutes to relax the gluten. When you return to the dough, it should stretch out much easier.
Spread the softened butter all over the dough. The softer the butter is, the easier it is to spread in this step. (Microwave it for a few seconds to soften if needed.) In a small bowl, mix the sugar and cinnamon together. Sprinkle evenly over the butter. Tightly roll up the dough to form an 18-inch-long log. If some filling spills out, sprinkle it on top of the roll. With an extra sharp knife, cut into 12 even rolls, about 1.5 inches each. Arrange in the prepared baking pan.
Cover the rolls tightly and allow to rise until doubled in size, about 1 hour. (Or use the overnight option below.)
Preheat the oven to 350°F (177°C). Bake for about 25-28 minutes or until they are lightly browned on top. After about 15 minutes, tent a piece of aluminum foil over the top of the pan to prevent the tops from browning too quickly and baking unevenly. Remove pan from the oven and place pan on a wire rack for about 10 minutes as you make the icing.
In a medium bowl using a handheld or stand mixer fitted with a paddle or whisk 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 1 minute. Spread the icing over the warm rolls and serve immediately.
Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 2 days or in the refrigerator for up to 5 days.