The sentence 'My favourite vegetable is peas' is not - strictly speaking - incorrect. But it is VERY odd. The fact is that no native English speaker would say this because it is simply not a natural-sounding sentence. The mismatch between the singular verb 'is' and plural noun 'peas' sounds very wrong to the native-speaker ear.
A much more natural phrasing is this:
Peas are my favourite vegetable.
Cats are my favourite pet.
Similar constructions that also sound fine to us are the novel title 'Oranges are not the only fruit' and the song title 'Diamonds are a girl's best friend'. By contrast, 'A girl's best friend are diamonds' sounds very wrong indeed.
Here's some advice - don't fret about sentences which simply aren't used in English, like 'My favourite vegetable is carrots'. Forget about them. Instead, concentrate on sentences which we do use, i.e. 'Carrots are my favourite vegetable'.
Now, here's a question for you. Why are Chinese students so obsessed with this particular sentence? Every few weeks, a Chinese italki member comes up with the sentences 'My favourite fruit is apple' or 'Apples is my favourite fruit', insisting that they are correct or wanting to know why they're incorrect. Does this come from a textbook or course that you all follow, perhaps?