Proof the yeast: In a large bowl, whisk the warm milk and sugar. Sprinkle over the yeast. Let it stand 5 to 10 minutes, until foamy. If it doesn’t foam, the yeast may be old or the milk too hot. Start over. It happens.
Make the dough: Whisk in the eggs and melted butter. Add 4 cups flour and the salt. Stir with a sturdy spoon or mix on low with a dough hook until a shaggy dough forms. If very sticky, add more flour a tablespoon at a time. Knead by hand on a lightly floured surface or with the mixer on medium-low until smooth and elastic, about 8 to 10 minutes. The dough should be soft, slightly tacky, but not gluey.
First rise: Shape into a ball. Place in a lightly oiled bowl, cover, and let rise in a warm spot until doubled, about 60 to 90 minutes. A slightly warm oven (turned off) with the light on works in a pinch. I’ve also tucked the bowl near a sunny window… cozy and reliable.
Make the filling: In a small bowl, combine the soft butter, brown sugar, granulated sugar, cinnamon, and a pinch of salt. It should be spreadable and sandy-sweet.
Shape: Turn the dough onto a lightly floured counter. Roll into a rectangle about 16 by 12 inches. Spread the filling evenly, leaving a small border on the long far edge.
Roll and cut: Starting from a long edge, roll the dough up tightly into a log. Pinch the seam to seal, then roll the log gently to even it out. Trim the very ends if they’re sparse, then cut into 12 even pieces. Dental floss (unflavored) slides under and up beautifully for clean cuts. A serrated knife also works with a light sawing motion.
Second rise: Arrange the rolls in a buttered 9 by 13 inch pan, leaving just a little space between each. Cover and let rise until puffy, 30 to 45 minutes. They should look relaxed and a touch larger, not blown out.
Bake: Preheat the oven to 350°F (175°C). Bake the rolls for 22 to 26 minutes, until golden on top and set in the center. If you own an instant-read thermometer, the middle should be around 190°F. If the tops are coloring too fast, tent loosely with foil.
Make the icing: Beat the cream cheese and butter until smooth. Add powdered sugar, vanilla, milk, and a pinch of salt. Mix until creamy and spreadable. If you like a thinner glaze, add another splash of milk.
Finish: Let the rolls cool 10 minutes. Spread the icing over warm rolls so it sinks into the swirls. Serve just-warm. They’re lovely the day of, but leftovers reheat well.