In a large bowl, combine room temperature milk, sourdough starter, and sugar until well combined. There should be no clumps remaining from the sourdough starter.
Add the flour and salt.
If using a stand mixer, mix the dough for 5-7 minutes on low speed until the dough pulls away from the sides of the bowl (see note 1). If kneading by hand, knead for 8-10 minutes until the dough passes the window pane test (see note 2). Dough should be smooth and easily stretchable.
Transfer your dough to a large, greased bowl and cover it with plastic wrap or a damp dish towel.
Bulk ferment. Allow the dough to rise for 8 to 12 hours, or until it has doubled in size (see note 3).
When the dough has doubled in size, punch it down and turn it out onto a clean, lightly floured surface. Flatten the dough so it is about 1 inch thick. Scatter ⅔ of the sun-dried tomatoes and mozzarella cheese evenly across the bread.
Fold the sides up and over in thirds, covering the cheese and tomatoes in the process. Add the remaining cheese and tomatoes on top and roll the bottom edge up to the top edge, creating a ball.
Second rise. Transfer your dough to a well-floured banneton or medium-sized bowl with parchment paper. Set the dough aside to rise until it has doubled in size, about 2 hours.
Preheat the oven to 500 degrees Fahrenheit.
When the dough has doubled in size, place it in a parchment-lined Dutch oven, cover with the lid, and transfer to the oven. Bake for 10 minutes, then lower the oven temperature to 400 degrees Fahrenheit and continue baking for 25 minutes. Then, remove the lid of the Dutch oven and continue baking for 15 minutes.
Allow the bread to rest for at least one hour before cutting. Enjoy!