Create a new branch for your work, and checkout to that branch
write contents in a new file (e.g. chapter1.md).
add an entry in _toc.yml (e.g. - file: markdown-notebooks)
Make a new PR to merge into the main branch.
(for publisher/administrator) after the working branch is merged, execute jupyter-book build . and then ghp-import -n -p -f _build/html. ghp-import will automatically push all the changes into the gh-pages branch. GitHub Actions will do the rest.