In the bowl of a stand mixer fitted with the paddle attachment, beat warm milk, sourdough starter, granulated sugar, melted butter, sour cream, and egg at low speed just until combined.
In a large bowl, whisk together 3 cups (417 grams) flour and salt. Stir half of flour mixture into milk mixture by hand. With mixer on low speed, add remaining flour mixture, beating until combined.
Switch to the dough hook attachment. Beat at medium speed until smooth and elastic, 4 to 8 minutes; add up to remaining 1/3 cup (42 grams) flour, 1 tablespoon (8 grams) at a time, if needed. (Dough should pass the windowpane test; see opposite page) Turn out dough, and shape into a smooth ball.
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 3 hours.
Spray a 13×9-inch baking pan with cooking spray.
Turn out dough onto a lightly floured surface, and roll into an 18×12-inch rectangle.
In a small bowl, stir together brown sugar, ½ cup plus 2 tablespoons (141 grams) softened butter, and cinnamon. Spread mixture onto dough, leaving a ¾-inch border on one long side. Starting with long side opposite border, roll up dough into a log. Trim ends. Using unflavored dental floss, score dough into 12 slices (about 1½ inches each); place floss under log, lining up with scores, and quickly cross ends of floss to create an even cut. (Alternatively, using a serrated knife, cut log into 12 slices [about 1½ inches each].) Tuck ends under rolls, and place rolls, tucked end down, in prepared pan. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed and rolls are touching, about 3 hours. (See Note.)
Preheat oven to 375°F (190°C).
Pour warm cream onto rolls.
Bake until golden brown and an instant-read thermometer inserted in center registers 190°F (88°C), about 25 minutes. Let cool in pan for 10 minutes.
In the bowl of a stand mixer fitted with the paddle attachment, beat cream cheese and remaining 1½ tablespoons (21 grams) softened butter at medium speed until creamy, 2 to 3 minutes. With mixer on low speed, gradually add confectioners’ sugar, beating until mixture is fluffy. Stir in room temperature milk until combined. Spread icing onto warm rolls.