Whisk the warm milk, yeast, and 1 tablespoon of the sugar in the large bowl. Wait 5 minutes, until a foamy cap forms.

Whisk in the remaining sugar, melted butter, egg, and salt. Stir in 3 cups of the flour until the dough looks shaggy with no dry streaks.

Knead on a floured counter for 8 minutes (6 minutes on medium low with a dough hook), adding the last 1/4 cup flour by the tablespoon only if needed, until the dough looks smooth and springs back slowly.

Set the dough in the greased bowl, cover with plastic wrap, and let it rise at 75 to 80°F for 60 to 90 minutes, until doubled.

Roll the dough into a 16 by 12-inch rectangle about 1/4 inch thick, until the corners look square.

Spread the soft butter over the dough, leaving a 1/2-inch bare strip along one long edge. Sprinkle on the brown sugar mixture, until the surface looks dark and shiny.

Roll the dough tightly toward the bare strip, pinch the seam, and cut 12 rolls about 1 1/3 inches wide with floss, until the spirals look tidy.

Arrange the rolls in 3 rows of 4 in the greased pan, cover loosely, and let them rise 30 to 45 minutes, until puffy and touching. Heat the oven to 350°F with a rack in the center about 20 minutes before this rise ends.

Bake in the center of the oven for 22 to 26 minutes, until the tops turn light golden and a thermometer in a middle roll reads 190°F. If the tops darken early, tent with foil and bake 3 to 5 minutes longer.

While the rolls bake, beat the cream cheese and butter for 2 minutes. Add the powdered sugar, vanilla, and milk, until the icing looks thick and glossy.

Cool the pan for 10 minutes, then spread the icing over the warm rolls, until it softens at the edges and sinks into the swirls.