Warm the milk to about 100°F (38°C). In a large bowl, stir it together with the melted butter, granulated sugar, sourdough discard, and egg until smooth.

Add 3 cups of flour and the salt, mixing to a shaggy dough. Knead on a floured surface for 8-10 minutes, adding up to 1/2 cup more flour, until smooth and springy. Let rise until doubled.

Punch down the dough and roll it into a 16x12-inch rectangle. Spread the 3 tablespoons soft butter over the surface, leaving a 1/2-inch border on one long side. Sprinkle evenly with the brown sugar and cinnamon.

Roll up tightly from the opposite long side and pinch the seam to seal. Cut into 9 equal pieces and place cut side up in a 9-inch parchment-lined round pan.

Cover and let rise until doubled and touching, about 1 1/2 to 2 hours (or refrigerate overnight, then finish rising 45-60 minutes at room temperature).

Heat oven to 375°F (190°C). Bake for 22-25 minutes until deep golden and the center reads 190°F (88°C). Cool in the pan for 10 minutes.

Beat the cream cheese, powdered sugar, 2 tablespoons milk, and vanilla until smooth and pourable. Spoon over the warm rolls and serve.