Ask HN: How to communicate adaptability to employers?

In looking for software engineering work over the last few months, I've found many openings for which I would be qualified, if given 2-4 weeks of training. In this economic climate, though, employers can find someone who is already experienced in the exact ways that they want, and they tend to go with the "safe bet".

I suspect many HNers are similarly frustrated.

As an aspiring generalist, I've explored many avenues in computer science. If I get the chance to sit face-to-face with someone, I am 100% able to communicate my often deep knowledge about all sorts of technical subjects--especially machine learning and functional programming. But I have no "proof" of my expertise on these subjects, and therein lies my conundrum.

How can I communicate that, although I may not "look good on paper", I learn and adapt to new situations and technologies very quickly, and am worth interviewing?