I'm not sure what topics are covered by 'electronics graduate.' Did this cover things like digital design/logic design?
I should make a list of topics from the absolute basics like binary arithmetic through more complex things like sequential logic and state machines.
The job aspect is more complicated for me. In the areas I've worked, a 4 year bachelors in CS, CE or EE is a prerequisite. Sometimes a masters degree as well, although that has varied over the years.
Startups are, I think, more likely to bring someone in based on their public work. I don't have startup experience so I'm not the right person to answer that beyond speculating.
If I were founding a startup and needed FPGA talent, I'd be open to interviewing someone that could demonstrate understanding of clock domains, synchronization schemes, state machine design, pipelining, AMBA. If you had demonstrated board design/prototyping, even better.
One of the projects I would have liked to have done in the past personally would be to make an effects processor for an electric guitar/bass. Really, even without any signal processing, this would be an impressive project. Making use of analog-to-digital converter(s), bringing in the data to the FPGA, having a place holder block that just passed the data through, and an interface piping data to a digital-to-analog converter would be a useful platform for demonstrating digital design understanding.
And really this could apply to any sort of "signal." Guitar just happens to be a place where digital logic and signal processing intersects nicely. Microphone and voice would work too.
You could really go nuts with this though, use it as a platform to learn about filtering, frequency shifting. You could make a harmonizer. Come up with your own auto-tune. It would be fun with immediate feedback.