Chinese Yams

Can portions of this year's root growth be planted for next year?

There are several different species called "Chinese yam". The one we have is known as "cinnamon yam" or "Chinese wild yam". The botanical name is Dioscorea oppositifolia.
Yes, it is possible to cut or break the tubers into pieces for planting. Portions of the tubers cut like a potato with "eyes" can be planted.
The usual method of propagating the plant, however, is to plant the aerial bulbils that form along the lengths of the vines in summer. These are anywhere from 1/4 inch to 3/4 inch in diameter. Using the bulbils allows one to leave the tubers in the ground intact, which enhances winter survival where the plant is hardy or allows the whole tubers to be harvested for use.