In a small bowl, combine 1 cup (240 grams) warm milk and yeast. Let stand until foamy, about 10 minutes.

In the bowl of a stand mixer fitted with the paddle attachment, beat brown sugar, melted butter, sour cream, egg, and remaining ¼ cup (60 grams) warm milk at low speed just until combined.

In a large bowl, whisk together 4 cups (500 grams) flour and salt. Stir half of flour mixture into sugar mixture. With mixer on low speed, add yeast mixture, beating just until combined. Beat in remaining flour mixture.

Switch to the dough hook attachment. Beat at low speed until smooth and elastic, 8 to 10 minutes; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if needed. (Dough should pass the windowpane test; see Notes.) Turn out dough onto a lightly floured surface, and shape into a smooth ball.

Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. (See Notes.) Loosely cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.

Between 2 sheets of parchment paper, roll Chocolate Chip Cookie Filling into a 16½×10½-inch rectangle (about ¼-inch thickness). Transfer filling between parchment to refrigerator. Refrigerate until slightly chilled but still malleable, 15 to 20 minutes.

Spray a 13×9-inch baking pan with cooking spray.

Lightly punch down dough. Cover and let stand for 5 minutes. Turn out dough onto a lightly floured surface, and roll into an 18×12-inch rectangle.

Remove top parchment from chilled filling, and carefully invert onto center of dough, leaving a ¾-inch border on all sides. Starting on one long side, roll up dough, jelly roll style. Using a serrated knife dipped in flour, cut into 12 slices (about 1½ inches each). Tuck ends under, and place, 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, 30 to 45 minutes.

Preheat oven to 350°F (180°C).

Using a pastry brush, brush warm cream over rolls; pour any remaining cream around edges of pan.

Bake until tops are golden brown and an instant-read thermometer inserted in center registers 190°F (88°C) to 200°F (93°C), 25 to 30 minutes. Let cool in pan for 10 minutes.

In the bowl of a stand mixer fitted with the paddle attachment, beat cream cheese at medium speed until creamy, 4 to 5 minutes. With mixer on low speed, gradually add confectioners’ sugar, beating until fluffy. Stir in room temperature milk until combined. Spread icing onto warm rolls.