In a large bowl, whisk together all purpose flour, bread flour, and salt.
Add blenderized tomatoes, sourdough starter, and olive oil. Mix just until a shaggy dough forms. Cover loosely with a towel or plastic wrap and let sit for 20-30 minutes.
With wet hands, stretch and fold the sides of the dough into the middle until the dough becomes a fairly smooth ball. Avoid overworking the dough, stop once all the remaining dry bits are incorporated.
Transfer dough to a clear bowl or container. Mark the position of the top of the dough with a washable marker or rubber band. Cover loosely with plastic wrap or a damp towel and proof until doubled in volume, about 8-12 hours (see note).
Grease the bottom of a 9x13 inch pan with about 1-2 tbsps olive oil, or generously butter a casserole dish. Transfer dough to pan. With wet hands, press your fingers into the dough repeatedly, starting from the middle and working your way out until the dough reaches the edges of the pan. If dough starts to resist further stretching, stop and allow the dough to rest for 5 minutes before stretching further.
Cover with lightly greased plastic wrap or a towel and allow to rise until puffy and about doubled in volume, around 2-4 hours.
Preheat oven to 400°. Cover the top of the dough with about 1-2 tbsps olive oil. Shove halved cherry tomatoes into the dough. Sprinkle the top of the dough with rosemary and parmesan cheese.
Bake for about 30-40 minutes or until golden brown and internal temperature of at least 200°. Allow to cool in pan for 10 minutes then remove to a wire rack to cool completely.
Focaccia is best enjoyed within 1-2 days of baking. Alternatively, it can be frozen in an airtight bag, and thawed in the fridge or at room temperature.