There are some situations that can give you the necessary insight (such as, if you already work there, or know someone who does, or you work for a supplier, customer, or competitor). In general, though, I think you're right that it's hard to just pick an employer and then figure out what they need.

In practice, you're probably better off to come at it from the opposite direction:

• What would you do for a living, if you could do anything?
• What can you do well that people will pay for?
• If you could replace a few of your current tasks that you don't like with ones that you do, what would your job look like?

Think about the answers to that sort of question first, then look around for companies that could benefit from your skills and expertise.

Beyond that, probably networking is the key. Get to know some of the people at the companies that you're interested in. Listen when they say things like, "My manager doesn't understand X" or "The factory always screws up Y." (They might even say, "We can't find anybody to do Z," although you can't really expect that.) Do enough of it, and you'll eventually find a match--a problem where the solution is you.