"Especially dumpling" is a whole new clause in addition to "I like Chinese food". In this clause, the subject and verb are elided:
I like Chinese food; I especially like (Chinese) dumpling.
So it is an elided clause. As such, we can regard it as not having a phrase structure category according to a grammar. The basic hypothesis behind elision is that there is a grammatically complete utterance in the speaker's mind, from which words are deleted. The original utterance obeys the rules of grammar and can be identified as some phrasal category like "clause". The version which is actually spoken, with removed words cannot.
For instance, what part of speech is "John Mary" in the following:
Bob is going with Jennifer and John with Mary.
it doesn't have one; if we accept the elision hypothesis (which we practically must), the speaker produced "John is going with Mary", which is a clause, and condensed it to "John Mary", which we can identify with that clause, and call it an "elided clause".
Note that "dumpling" is normally countable, but here "dumpling" is uncountable. This is a little bit strange, but it will be interpreted as being similar to "I like duck".
Of course "duck" has to be treated uncountably when we are talking about food because it is short for "duck meat"; "I like ducks" cannot be used when talking about food, whereas "I like dumplings" is the normal way to state your preference for steamed or boiled dough.
"For example" patterns can also be hypothesized as being cases of elision. Consider that a sentence like:
I like Chinese food, for example: pork dumplings, won ton soup and barbecued duck.
can be viewed as a condensed version of this string of clauses:
I like Chinese food. For example, I like pork dumplings, won ton soup and barbecued duck.
For example is a kind of introductory interjection, which indicates that the following clause gives examples. The list of foods is then a compound noun which serves as the object of the elided verb "to like", where the elided subject is "I".