Oh goats and chickens, that's a hard one. Nubian and Boers are pretty big and an australorp can probably fit thru a 10x12 door.
I've read that if you use a V shaped opening(oriented just like the letter V) just big enough for the chickens, the goats can't bring their feet to the centerline of their bodies to pass thru a V shaped opening even if their bodies are small enough to get thru. Haven't ever seen it implemented but would love for you to test the theory.
Or you can use an L shaped(from the top) tunnel type pass thru. 10 x 12 tunnel(2 sides and a top) that goes out about a foot then a 90 degree turn and out another foot...the goats can't make the tight turn but the chickens can. Could be made out of solid wood and/or 1x2 mesh.