Place mushrooms in a medium saucepan. Add enough water to cover by 1 inch. Bring to a boil. Reduce to a simmer, and cook until tender, about 30 minutes. Drain, and coarsely chop; set aside.
Meanwhile, place tomatoes in a large saucepan. Bring to a boil. Reduce to a simmer, and cook for 20 minutes. Add porcinis, and simmer until desired consistency, about 30 minutes. Add butter, cheese, and basil; simmer for 5 minutes.