1. Combine hearts and gizzards with water and 1 teaspoon salt, and simmer 15 minutes. Drain and remove membranes from gizzards, and chop gizzards and hearts fine.
2. Saute onion, celery and carrot in olive oil until lightly browned. Add tomatoes, mushrooms, herbs, pepper, remaining 1 teaspoon salt and the chopped giblets.
3. Bring to a boil, cover and simmer for 1 hour.
4. Cook rigatoni in boiling salted water as package directs. Drain, and toss with butter and cheese. Add the giblet sauce and mix lightly.