The short answer is that no, I'm afraid you can't do that. But I think that examining the background to this might help to explain what at least some of the issues are:
One of the things about Audition that you have to remember is that its history doesn't start with Adobe - and because it's always had a core following (used by just about every radio station in the world as Cool Edit Pro before Adobe bought Syntrillium, the company that created it) it has been very difficult to make major structural changes to it, as they would cause a degree of chaos amongst the core of users who are used to it just the way it is. What this has meant in practice is that changes have been introduced to it very slowly, and really, nothing major has changed structurally - so far.
And as such, it's never used the 'project' model that Premiere has as an overarching feature, which is what you are seeking. If you want to do that, you have to create your own project folder, but it won't be the same - you'll still have to open each session file separately. Audition is perfectly happy having several different sessions available and open at the same time, and you can jump back and forth between them, but that's as far as it goes - certainly at present.
But you have to ask some other questions too. Is there really any point in trying to make Audition look and behave like Premiere? Why would you need both? Without going into the ins and outs of it, Adobe clearly think that there's a demand for a separate audio editor, and to this end they've gone for a 'round trip' model of processing as far as Premiere is concerned.
Is it reasonable, one might ask, for video users to try to dictate the way that Audition works, to make it more like Premiere? Well a lot of video users seem to think it is, and to be fair, some of the ideas they've come up with certainly have a degree of merit to them - I'm not denying that. One that's cropped up quite regularly is the one about bins, which Audition doesn't have. A fine idea in itself, although in practice it's not that simple with audio, because there are no pictures to put in them, so they'd all look the same. Nevertheless, I believe it's being looked into. Now if that came to any degree of fruition, then it would also perhaps be a good time to introduce a project structure. Personally I wouldn't have a lot of use for it but I can see some audio users thinking that it would be wonderful for them and their workflows.
What to do? My suggestion would be to write it up and submit it on the Audition ideas forum where the staff look quite regularly, which they don't do here.
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