Which side of an over-tightened brass NPT joint needs to be replaced?

It is not certain that you have damaged the joint. Take it apart and reassemble with pipe joint compound. Some people use both Teflon tape and pipe joint compound. To do this I would apply a small amount of joint compound to the male threads, then wrap with tape.

It used to be one only applied pipe dope to the male threads, but I had a case recently where I had to redo a joint three times because of seeping. Finally I applied joint compound to the female threads too. I attribute these problems to design/manufacturing of the threads being different than in the past.

The next time I have to do one of these I intend to try "plumber's hemp" video which is used in Europe.

Five years ago I redid the clothes washer connections with Teflon tape and they seeped. Tightening did not stop the seep. I redid it with joint compound and it still seeped, but much less. I needed to get the washer in service so I wrapped a rag around the joint to absorb the seep and hoped it would stop. I looked after a few days and no seep.

A year ago I installed new outside valves. Our 1970 tract house has two outside taps on 1/2" copper tubing coming out of the ground with the taps originally sweated directly to the copper. In 1980 I redid them after they were damaged in a freeze, sweating on a 1/2" FIP adaptor.

This time I selected quarter turn 3/4" ball valves, labelled garden valves. I think ball valves are not really the best choice because they are really not metering valves, just on-off, but time will tell. This required two intermediate fittings to go from the 1/2" FIP fitting on the copper tube to the 3/4" male on the valve. I had to redo the joints several times to stop leaks, and I left it seeping. After a few days the seeping stopped.

In the store this Italian made "garden valve" was easy to turn even though it has only a small handle, round with a single projecting tab, but when installed and pressurized they were hard to turn . I could operate them by using both hands but my wife could not. I looked into replacement handles, but couldn't find any, then realized that an extension handle would work.

A 7 inch long extension handle of 1/2" hard copper with a splice connector soldered on the end worked. This slips over the tab on the handle. For the valve that my wife uses frequently I made a "wrench" to operate the valve without bending over. This is a 2 ft length of 1.5" dia PVC tubing with a notch in the end to engage the single tab on the valve handle.

EDIT

I would think that "hemp" is not specifically required. I would try heavy sewing thread. Put pipe "dope" on male threads, then wrap many turns with thread so that the thread does not get swept into the water stream. Be an interesting experiment.

More EDIT

I think US NPT threaded fittings used to be tighter than they are now. It seems to me that resistance would be encountered after one or two turns. But when I was selecting fittings at HD from their extensive set of brass NPT fittings they seemed looser than I remembered. I concluded quality has slipped when manufacturing went outside the US, but there is an alternative explanation.

It could be we are now getting European standard / world standard threaded fittings for plumbing and these might have always been looser and which required pipe dope and hemp or equivalent fiber. Fiber doesn't sound inert, it could be that microorganisms could grow there, but I have not heard of any such problems.

Still more edit There are US companies which advertise precision NPT threaded fittings which seal with Teflon tape alone, e.g., Hoke catalog

yet more editing Now this is real plumbing video of plumber threading steel pipe

Try this; take the fitting back apart. Put 2-3 wraps of teflon tape (no more than 3). then coat the male threads with RTV silicone caulking. snug as normal and let set overnight. I would normally not use the teflon tape but the brass has been stretched and enlarged by the over tightening. When I as still working I would put whole jobs together using only RTV silicone and no pipe dope or teflon tape.

Probably poor quality control in the manufacturing of one or more of your parts. Looking at all the remedies I see here posted, if you either want to put more "seal" on threads / and/ or start anew with new parts, whichever is cheaper, of which I'd say more sealant.