Stir together yeast, warm water, and 1 teaspoon of the granulated sugar in the bowl of a heavy-duty stand mixer fitted with paddle attachment. Let stand until foamy, about 5 minutes. Add eggs, 1 cup of the milk, 1/2 cup of the granulated sugar, 1/2 cup of the butter, 2 cups of the flour, and 1 teaspoon of the salt to bowl. Beat on low speed just until dough comes together, about 1 minute. Replace paddle attachment with dough hook, and add 2 cups of the flour to bowl. Beat on low speed until dough is smooth but still very sticky, about 5 minutes. Turn dough out onto a surface floured with remaining 1 cup flour, and knead until elastic and almost no stickiness remains and all flour is incorporated. Transfer dough to a lightly greased large bowl, turning once to grease top. Cover with a damp towel, and let rise in a warm place (80°F to 85°F) until doubled in size, about 1 hour.
Stir together brown sugar, cinnamon, 1/2 teaspoon of the salt, and remaining 1/2 cup granulated sugar in a small bowl. Turn dough out onto a lightly floured surface, and roll out into a 20- x 14-inch rectangle. Spread 1/2 cup of the softened butter over top of dough rectangle; sprinkle with brown sugar mixture. Starting at 1 long end, roll dough into a long rope, stretching dough back to a 20-inch length, if necessary. Starting in center of a 12-inch cast-iron skillet, wrap dough rope into a tight coil, working outward. Using a sharp paring knife, cut slits through top layers of dough at 1-inch intervals. Cover with plastic wrap, and chill 8 hours or overnight.
Place a large bowl of very hot water in oven, and let stand 5 minutes. Place skillet with chilled dough in oven, and let rise until doubled in size, about 1 hour. Remove skillet and bowl of water from oven. Preheat oven to 350°F.
Place skillet on an aluminum foil-lined baking sheet, and bake in preheated oven until deep golden brown, about 30 minutes. Cover with foil, and bake until dough in center is cooked through and a thermometer inserted in center registers 190°F, 20 to 30 minutes. Cool in skillet 10 minutes before cutting.
Meanwhile, beat cream cheese and remaining 1/4 cup butter with an electric mixer on medium-high speed until fluffy. Add powdered sugar and remaining 2 tablespoons milk and 1/4 teaspoon salt, and beat until smooth. Drizzle icing over warm cinnamon roll. Serve warm or at room temperature.