Nonsense.
The JEE spec has lots of "should nots" and "thou shant's". The Servlet spec, on the other hand, has none of that. The Servlet spec is much more wild west. It really doesn't dive in to the actual operational aspects like the JEE spec does.
I've yet to see a JEE container (either a pure servlet container ala Tomcat/Jetty, or full boat ala Glassfish/JBoss) that actually prevented me from firing off a thread on my own. WebSphere might, it's supposed to be rather notorious, but I've not used WebSphere.
If the concept of creating unruly, self-managed threads makes you itch, then the full JEE containers internally have a formal "WorkManager" that can be used to peel threads off of. They just all expose them in different ways. That's the more "by the book-ish" mechanism for getting a thread.
But, frankly, I wouldn't bother. You'll likely have more success using the Executors out of the standard class library. If you saturate your system with too many threads and everything gets out of hand, well, that's on you. Don't Do That(tm).
As to whether an async solution is even appropriate, I'll punt on that. It's not clear from your post whether it is or not. But your question was about threads and Servlets.
Just Do It. Be aware it "may not be portable", do it right (use an Executor), take responsibility for it, and the container won't be the wiser, nor care.