There are many discussions on this, for example this answer and this discussion. The short answer: both are correct, with "is" being the more common choice.
Subject clauses beginning with "what" are singular, as you said, we can substitute "what" with "the thing that". The verb agrees with the subject, and not with the object. That is why what matters is the "what-clause", as you call it, and not the predicate, and the verb will usually be "is" and not "are".
When the object is singular, it is quite clear:
What really matters is your health.
What I want is my own home.
Sometimes, however, when the object is plural, we want to emphasize that. That is when "are" is possible:
What really matters is/are the children and their needs, not our desires and our peace and quiet.
What we want is/are heroes.
Most of the time the choice would still be "is" but "are" would not be a mistake.