Try to run jekyll build in test with nox, mamba and aggressive caching

> So I was enthusiastic about using common nox command here at first, but I am no longer convinced. The old docker-based approach was building in 51 seconds, the nox/conda takes 6 minutes 24 seconds. It is a substantial delay. I can go for mamba, cache etc but it will all increase complexity and maintenance burden - caching is hard.

_Originally posted by @krassowski in https://github.com/jupyter/jupyter.github.io/issues/579#issuecomment-1002777932_

> Huh, that's surprising. But if it's that big a difference I agree. We could probably optimize the nox build down but that's not the best use of time in this PR in my opinion, unless you're enthusiastic to try it out

_Originally posted by @choldgraf in https://github.com/jupyter/jupyter.github.io/issues/579#issuecomment-1002782295_

> I guess it comes down to having the docker image already available on GitHub infrastructure (the docker image is the one from the action suggested by GitHub) vs having to resolve dependencies and install them from conda - I would not expect a big difference in the actual build step.

_Originally posted by @krassowski in https://github.com/jupyter/jupyter.github.io/issues/579#issuecomment-1002783348_

> Yeah for sure - maybe with mamba it'd be significantly faster? This also seems like it'd be a one time thing I'd we cached it properly. Environments all get stored in a .nox folder so maybe we could just cache that?

_Originally posted by @choldgraf in https://github.com/jupyter/jupyter.github.io/issues/579#issuecomment-1002786076_