Rinse giblets and neck. Wrap liver airtight and chill to use in gravy, or save for other uses.
In a 5- to 6-quart pan over high heat, melt butter; add gizzard, heart, and neck and turn often until well browned, about 5 minutes. Add carrots, celery, and onions and stir often until onions are lightly brown, about 5 minutes. Add broth and stir to release browned bits. Cover and bring to a boil. Reduce heat and simmer until gizzard is very tender when pierced, 1 1/4 to 1 1/2 hours.
Pour the broth through a fine strainer into a 2-quart measure. Reserve neck and giblets; discard vegetables. If you have more than 8 cups of broth, boil to reduce; if you have less, add chicken broth.
Nutritional analysis per cup.