BRP (Basic Role-Playing), the system Call of Cthulhu, Elric!, RuneQuest, and a number of other games use (afaik, as I have actual experience mostly with CoC only, but with that, a real lot) is a system in which successful skill use may lead to the development of said skill. It might fit your needs and requirements.
Edit: Of course, even a CoC character starts out with a number of pre-selected and therefore better developed skills that most likely match the character's profession / role in life, but practically all skills are available to anyone, only it's hard to achieve successful use with the "underdeveloped" ones. (Like, if your character is a carpenter, for example, getting a chance to use - let alone successfully use - your surgery skill will be quite a challenge. :)) However, it would be very easy to tweak the system to allow a wider range of starting skills (with lower scores) that would provide a better chance for the character to develop into a more specific "role" later.