I have the same issue - erosion and chickens wanting to eat everything poking up out of the soil. I begin by fencing off the area with deer netting to protect the new shoots until they get several feet high and less vulnerable to being dug up or completely eaten. I sow seeds in early spring and sometimes even in the fall. They have a better chance of getting established before the summer heat hits.
I use straight tree branches and 2x 4s laid perpendicular to the slope to hold soil in place during heavy rain and to help rain sink in rather than run off too quickly.
I planted alfalfa seeds, bee plant, rhudbeckia, cone flower, chard and spinach. Many of these will reseed each year. The chickens won't have access until the plants are a foot tall or more.