I haven't been able to find any documentation on how to implement that kind of chunked response on Cloud Run. That could be an intended behavior of how Cloud Run handles responses in the first place. Additionally, in the case of chunked transfer encoding, the Google Front End (GFE) is configured to remove the Transfer-Encoding header from responses served by Cloud Run applications, as the GFE itself buffers the response data before forwarding it to the client.

You could also consider submitting an issue report to Google's public issue tracker for Cloud Run. This would allow the Cloud Run engineering team to also look into it and potentially offer alternative solutions. Be aware, however, that there is no guaranteed timeline for a response.

You can also take a look at this Google Cloud blog about Cloud Run's support for streaming.