How to Make It
Whisk 1/2 cup water and 3 tablespoons flour in a small microwave-safe bowl. Microwave on high until mixture thickens to a paste, about 1 minute, stirring halfway through; set aside. Whisk sugar, yeast, salt, and remaining 3 and 3/4 cups flour in a large bowl. Heat milk and butter in a medium saucepan over medium-low, undisturbed, until butter melts, about 5 minutes. Remove from heat; let cool for about 5 minutes.
Add warm milk mixture, flour paste, and eggs to flour-yeast mixture; stir with a wooden spoon until a shaggy dough forms. Turn dough out onto a lightly floured surface; knead until smooth, 5 to 6 minutes, dusting surface with more flour if sticky. Wipe bowl clean and lightly coat with butter. Place dough in bowl and turn to coat. Cover with plastic wrap; let rise in a warm place until doubled, about 1 hours.
FILLING AND ICING
Grease a 13-by-9-inch baking dish with butter. Turn dough out onto a lightly floured surface. Roll into a 16-by-11-inch rectangle. Stir together butter, brown sugar, granulated sugar, cinnamon, and salt in a medium bowl; spread evenly to edges of dough. Roll into a log lengthwise and cut crosswise into 12 2-inch pieces. Place rolls in prepared baking dish about 1/2 inch apart. Cover loosely with plastic wrap; let rise until rolls are puffed and touching, about 1 hour.
Preheat oven to 375°F. Uncover rolls and bake until golden, 25 to 30 minutes. Let cool for 5 minutes.
Stir together powdered sugar and cream cheese in a medium bowl until combined. Whisk in milk until smooth. Spread icing over warm rolls.