What are some of the best ways to solve a chicken and egg problem?

There is no chicken <> egg problem.

Here's why:

1. In nature, the egg came first (both for all eggs, and chicken eggs).
2. As a metaphor, it's used to represent a paradox. A paradox cannot be logically true and correct.

The Wikipedia page on paradoxes explains it best:

Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, and require extending the context or language in order to lose their paradoxical quality.

To solve a situation in business which seems paradoxical, extend your context. Don't work within limits as restrictions, work around them. Use the tools you do have to eventually come to the conclusion you're trying to rush to right now.