Prepare the dough: In a large mixing bowl, whisk together warm milk, chai tea, sourdough starter, eggs, melted butter, sugar, and salt until smooth. Stir in about 3 cups of bread flour until a shaggy dough forms. Cover and let rest for 30 minutes.
Knead until smooth: Gradually add the remaining flour, kneading until the dough is soft, elastic, and slightly tacky but not sticky. This takes about 6–8 minutes. Test with the windowpane method: stretch a small piece thin without tearing.
First rise (overnight): Place dough in a lightly oiled bowl, cover, and let rise at room temperature until about 30% larger. This slow fermentation usually takes 8–12 hours.
Mix the filling: In a small bowl, combine softened butter, granulated sugar, brown sugar, and spices until spreadable.
Roll + fill: On a lightly floured surface, roll dough into a 12x18-inch rectangle. Spread filling evenly, leaving a small border.
Shape the rolls: Starting from the long side, roll the dough into a snug log. Pinch seam to seal, then slice into 12 equal pieces. Arrange in a parchment-lined baking pan.
Second rise: Cover and let proof until puffy and nearly doubled, 45–90 minutes depending on room temp. Optional: pour ½ cup warm heavy cream around the rolls before baking for extra softness.
Bake: Bake at 350°F (175°C) for 25–30 minutes, until golden on top and centers reach 190°F.
Make the frosting: Beat cream cheese and butter until smooth. Add powdered sugar gradually, then mix in vanilla, chai tea, and cinnamon until creamy.
Frost + serve: Spread frosting generously over warm rolls so it melts slightly into the swirls. Serve cozy, gooey, and chai-spiced.