Carthay Circle is one you might need to plan further ahead for. On my last visit I tried a few weeks ahead of time but couldn't get anything for dinner on a Saturday.
On my upcoming visit I tried about a month ahead of time and it was close. I sent one email but never received a confirmation, so I sent a second the next day and was told "we never received your first one, and we don't have the time you wanted, so you'll have to come an hour earlier." But oddly enough, I received a reply to the first email shortly after that saying I was confirmed for the original time I wanted. I proceeded to write a few more emails triple checking the original time and then canceling the other one. LOL