Can I create minify files right after empty cache?

I have been having a problem with my CDN not properly getting minified JS files from my server. I think the problem is with w3tc taking too long to generate the minify files when the CDN wants them and a timeout triggering on my server. I can’t modify the timeout, so I want to know if I can have w3tc generate the minify files right after it empties the cache, instead of w3tc waiting until it gets the request from the CDN and generate the files then. I notice that the minify files don’t exist on my server until after the CDN requests them, and I want those files to exist on my server right after I empty the cache, before the CDN requests them.