Do noun clauses need subjects?

"Homework is what I hate to do." "Homework is what makes me sad."

In the first sentence, "what I hate the most" is clearly a noun clause. It functions as a subject complement. And it begins with a subordinating conjunction (what), has a subject (I), and a predicate (hate).

In the second sentence, is "what makes me sad" a noun clause? Is it a clause at all?

It starts with a subordinating conjunction (what) and has a predicate (makes me sad), but what is the subject? Does what function as both a subordinating conjunction and subject?

Please help! lol