In a large bowl or stand mixer, combine sourdough starter, warm milk, sugar, eggs, softened butter, and bread flour. Mix and knead until the dough is smooth and slightly tacky. Cover and let rise 4–6 hours at room temperature, or overnight in the fridge.
In a small bowl, mix together the softened butter, brown sugar, cinnamon, and nutmeg until smooth and spreadable. Set aside.
Roll the dough into a rectangle on a floured surface. Spread the filling evenly over the surface.
Roll the dough up tightly into a log. Slice into rolls using plain (unflavored) dental floss for clean swirls.
Grease a cast iron skillet and arrange the rolls with a bit of space between each one to allow room for rising.
Cover the skillet and let the rolls proof for 1–2 hours, or until they are soft, puffy, and nearly touching.
Bake at 375°F (190°C) for 20–25 minutes, until golden brown and bubbling. If browning too quickly, loosely tent with foil.
While the rolls are baking, beat together the cream cheese and butter until smooth. Add powdered sugar, vanilla extract, lemon juice, and lemon zest. Add a splash of milk or cream to thin if needed.
Spread the lemon cream cheese icing over the warm rolls and let it melt into every swirl.