I'm not sure if I just have diva girls, but they absolutely love pineapple sage and lemon balm, but won't eat it off the plant. They need me to hold it so they can rip off small bits. Or I have to rip it up and throw it on the ground. They love thyme and can easily eat it off the plant, so we have to put hardware cloth down or put them in pots until they're established. Same with the oregano. But the larger plants seem to be safe. They also don't seem to bother the rosemary and lavender. 
The main thing is to help protect the roots to make sure they don't scratch them up. We've created some grazing boards in the run for the delicate plants (lettuce, other fodder stuff).