My dream is becoming an English teacher is not a grammatically acceptable English sentence.
This is a licensing matter. That is, there are four clause-types which can act as Noun Phrases; but each verb or adjective or noun which takes an NP complement licenses—that is, accepts—only a subset of these types.
The noun dream licenses a somewhat different set as predicate NPs:
- that clauses: My dream is that I will become an English teacher.
- to-infinitive clauses: My dream is [for me] to become an English teacher.
The verb dream, however, licenses only one type:
- that clauses: I dream that I will become an English teacher.
You cannot say I dream to become an English teacher. On the other hand, you can use the verb dream with a prepositional phrase headed by of + gerund (the -ing form of the verb):
I dream of becoming an English teacher.
The noun dream also licenses this prepositional construction as a modifying clause:
My dream of becoming an English teacher will be dashed if I cannot master the pluperfect.
But gerund clauses are not licensed by the noun dream as predicate NPs. If you want to use a gerund NP you must back into it with this awkward construction:
My dream is one/that of becoming an English teacher.
My dream is becoming an English teacher is therefore simply nonsensical. It means, literally, that your dream is undergoing the process of being transformed into an English teacher!
Unhappily, there’s no way of predicting what constructions any given word will license; you have to learn this word by word.
You will find more discussion of licensing here, here, and here