I wouldn't characterize this as "automatic", you have to do a lot of manual work to support reconnection in most cases. You need to have a meaningful "event id" that can be resumed on a new connection/host somewhere else with the Last-Event-Id header. The plumbing for this event id is the trivial part IMO. The hard part is the server-side data synchronization, which is left as an exercise for the reader.
Also, God help you if your SSE APIs have side effects. If the API call is involved in a sequence of side-effecting steps then you'll enter a world of pain by using SSE. Use regular HTTP calls or WebSockets. (Mostly b/c there's no cancellation ack, so retries are often racy.)