I have a new pistol build with 10.5" barrel.  After the first 15 rounds, it will no longer extract.  .223 Wylde chamber with 1:8 twist.

After each additional shot, it became harder and harder to manually extract the casings.  it's up to the point where it takes a rod to get them out.  Tried brass ammo, zinc plated ammo, and even steel ammo.

Brushed out the firing chamber, but even dropping a spent case in and closing the bolt carrier make it really difficult to extract.  Any ideas?  Or just get the barrel replaced?

I'll take a stab at this.  

Who is the manufacturer of the barrel?  This certainly isn't normal, so you might want to contact them and tell them what you are experiencing.  

If you want to troubleshoot this yourself, we need to know more about what it is doing.  Are the cases just being left in the chamber, while the bolt operates?  Will the bolt lock back on an empty magazine after firing?  If so, your barrel might be drastically overgassed, and it is trying to extract the case before the pressure drops enough to do so.  The extractor just pulls over the case rim, because the pressure is too high to allow the case to be extracted.  

A heavy buffer may help if this is the case, or you might need an adjustable gas block.  An O ring behind the extractor may help as well.  

However, I think I would contact the manufacturer before I started making changes.  

OK first thing is to get a look at the brass.  Look specifically at the rim.  Has the extractor damaged it?  If so one of the following are likely your issue...

Be sure the chamber is clean.  Often they are not cleaned by the manufacturer and tooling remnants can be lodged in there.  If it gets worse that quickly that's where I'd start.

Pistols are notorius for short dwell times causing FTEs.  You need to increase the dwell time, and typically as stated above a heavier buffer is the fix.  If you need to slow it down more a full-auto carrier weighs more than the standard semi so that's another choice.  You can reduce the gas via an adjustable block, but it still doesn't affect dwell time.  The pigtail gas tubes can do this as they increase the distance the gas has to travel, but it means pulling stuff to install it, where the buffer is an easy out and in affair.  I'd go to an H3 and see where that gets you, but the pigtail tube is likely the cheapest option, so it's a choice of convenience vs $$$.