"Required to do" vs. "required doing"

Solution 1:

I think all answers and comments so far, including mine, are really missing one crucial point.

The big thing here is that "required reading" is a set phrase that we use when we talk about books mandatory for some purpose, and particularly when talking about mandatory books for a course.

That reason, more than anything else, is why "required reading" is preferred here, even though "required to be read" would be grammatical and make a certain amount of sense.

Solution 2:

Note that in

Your employer is required "to deduct" a certain amount from your salary …

required is a verb, and deduct is also a verb.

However, in

'Hamlet' is required "reading" (=must be read) for this course

required is an adjective to the gerund (behaving like a noun) reading.

It is not a case of merely adding -ing to a verb. Consider that deduct is a verb and reading is a noun.