Migrate the examples to `hatch`

[jtpio]

Problem

To follow the lead of the cookiecutter: https://github.com/jupyterlab/extension-cookiecutter-ts/pull/222

And streamline all the examples.

Proposed Solution

Maybe the upgrade script would work with this and help do most of the work.

Additional context

https://github.com/jupyterlab/extension-cookiecutter-ts/pull/222

https://blog.jupyter.org/packaging-for-jupyter-in-2022-c7be64c38926