I have a leaky exterior faucet where the plastic packing nut has broken and the threaded end is still inside the faucet. How can I remove the broken packing nut from the faucet?
I'm assuming this is in reference to an outdoor garden faucet. If you can't get the old packing out, replace the entire faucet. But first, make sure you don't have a dope of a builder that would solder the threads. Try to unscrew the fixture from the wall (shutoff the outside water first). If it unscrews, then all you need to do is pickup some pipe dope and a new faucet from the store. Clean the old threads, put some pipe dope on the threads, and screw on the new fixture.
If you can do it without damaging a washer (or if the washer can be replaced easily), you can sometimes take a large bladed flat screwdriver and heat it with a plumbing torch, then while it is still hot, firmly push it into the remaining plastic until it is embedded. You can then twist it out fairly easily.