It's not something that occurs very often, but it's not unheard of. Sometimes the ejector post breaks in a 1911 pistol. In this picture you can see that the ejector is completely missing.
The problem is a fairly simple fix. But it's going to require a drill press or mill / drill to precisely drill out the old broken ejector part.
When I have a chance to replace these ejectors, I always opt for the slightly oversized one. That way you can fit them up nicely and tune them to kick out the brass casings in a direction that you might want them to go. Also, when I drill out the hole, I like to measure the diameter of the front post first. This way I can drill the hole slightly smaller and get an interference fit. This simply means that I have to tap the ejector post into the hole to make sure it doesn't have any wiggle room to move. The ejector takes quite the beating and I don't want it to move at all.
You'll notice that the ejector hangs out a little bit over the back of the frame. You can trim or file this flush with the back of the gun. Color the metal with some cold blue to match the finish.
The next step is to cross drill the hole that takes a pin and holds the ejector in place. I tap the pin into the hole while everything is still clamped tightly in the drill press vice.
You have to go really slow here because it's easy to snap off a drill bit this tiny. This is something I don't think can be done with a hand drill.....
Once the slide is back on the pistol, I take a look at the face of the ejector and see if it protrudes far enough from the slide cut out. The only thing to do now is test fire the gun and see where the empty cases are being thrown. Are they clearing the ejection port fine, or does it throw cases into the shooters face ? A small touch up with a file on the face of the ejector changes this.
A final check on the back of the pistol to ensure that there's no visual problems evident. Also check the color of the metal you ground off and make sure it matches the finish of the gun.