We are moving to a bigger piece of land soon and will be adding goats. I have been running the same "how to feed chickens and goats separately in the same pen" thing through my head for a couple weeks now. I have really only come up with one decent idea besides what others mentioned here. The small doggy door type option for chickens only works if your goats can't fit through it. We are planning on Nigerian Dwarfs or Pygmies so that option is pretty much out for us too. Here is my un-tested grand idea:
1. Screw a wide, shallow, metal feed pan to a piece of plywood, pallet, or some sort of stand not easily toppled over by goats.
2. Cover the top of the pan with chicken wire.
3. Stretch wire tight and attach to wood surface
This could solve the problem of food waste by chickens scratching and keep the goats out of the feed. I have a sneaking suspicion that goats may be able to easily tear through the chicken wire though. Maybe a heavier gauge wire with small holes like chicken wire, if you could find it. If not, several layers of strategically placed heavier gauge wire fencing with larger holes. I think it could work!
Another way to keep the chickens from wasting their food is to ferment or soak it. There are fermented threads. I can't look them up right now.