Prepare levain by mixing sourdough starter with all-purpose or bread flour and water. Let sit for 10-12 hours at 78-80°F.
In a large bowl, mix ripe levain with water, salt, and bread flour until a wet, sticky dough forms. Cover and let rest for 30 minutes.
Perform a series of stretch and folds every 30 minutes for 1 ½ hours, incorporating cold butter into the dough.
Let the dough rise for 6-7 hours until it is about 70% risen and has bubbles visible.
Stretch the dough into a rectangle, spread cheese and ham, fold it over and roll it up.
Cover and refrigerate the shaped dough overnight or up to 12-18 hours for cold fermentation.
Preheat a Dutch oven to 450°F for 20 minutes.
Score the dough, sprinkle with remaining cheese, and place in the preheated Dutch oven.
Bake covered for 25 minutes, then uncover and bake for an additional 20 minutes at 425°F until bread is cooked through.
Let the bread cool on a baking rack before serving.
