I take it that “chicken and dumplings” is the name of a dish in which chicken is cooked with dumplings. In that case, the two nouns go together to form a single unit, and is treated as a singular noun.
Other examples are:
Toast, butter and jam is what I normally eat at breakfast.
Nasi lemak and sambal is delicious in the morning, but it makes us sleepy.
Fish and chips is a typically English dish, and tastes best when it iswrapped in paper.