Cooking messy foods in a bamboo steamer without a cleaning nightmare
My mom uses pieces of lettuce, cabbage, spinach or some other large, leafy green as a bottom for her steaming uses.  I can't explain why this works, but for dumplings and Shanghai-style dumplings, this prevents the pool-of-water below as you describe.  Perhaps it has something to do with the varied height with the leaves...offering the water pools a place to collect without staying in contact with the food.

This may or may not catch the chicken gunk you're referring to, but I think it's worth a shot.