I read all of them, even the greyed out ones, until I'd read enough to trust that the HR team's filter was well-calibrated, and I also let the HR team know when they got it right/wrong so they could improve their filtering process.
Overall it worked great but I understand that such functional HR teams are the exception, not the rule.
Edit: My apologies everyone. I actually read this as "cover letter" instead of CV.
I agree a resume (as noted below) should be required, but CV (academic style) is probably a US term not applicable here.
Sorry!
edit: saw another response which said a cv was a resume, this is correct. In the UK we use Curriculum Vitae (CV) instead of resume. They are the same thing. Not sure what you may have thought a CV was, however I would be interested to know.
I did exactly this, though I'm pretty sure I'm not your boss.
As a side note, HR's insistence on doing the initial resume review and phone screens usually skews the entire recruitment process. I've had situations where HR gave me 20 resumes and didn't even look at the rest.