how to remove frost free outdoor faucet drain valve assembly
That part is called the stem, and is typically fully replaced, or more often, you simply replace the o rings and, well, as noybman mentioned, the bushing(s).

I have a feeling, if you pull the stem completely out, you'll fine the little brass part you're really looking for. There's a base at the back of that stem that closes the valve, and it will be replaceable on some spigots, in whole or in part. If yours is replaceable, it is typically adjustable as well.

Do a google image search for your spigot under 'repair kit'. You'll get a very handy diagram.