Then give them a coding problem that takes a short amount of time but allows you to evaluate their skills. I relish these opportunities and respect the companies that ask me to do them.

To keep with your metaphor, ask the carpenter to make you a cabinet door.

Again, I agree with others here... if you're passionate about the startup you're working at (and you should be), that is your side project. I distrust people who work at startups that have side projects.