Add an example for FastAPI integration with UI modifications

[MA3CIN]

I'm trying to modify the chainlit UI (send a message from the agent) via a FastAPI endpoint, and am running into some issues (described here: Chainlit/chainlit#1356). The documentation is very sparse (https://docs.chainlit.io/integrations/fastapi) and doesn't include a single example with any chainlit functionality.

The cookbook also doesn't contain a FastAPI example that modifies the UI (as in displays a new message)

Documentation is also extremely sparse on init_http_context(), which seems to be the key here with FastAPI integration

Can i get a minimal working example?

[MA3CIN]

Bump

[nethi]

Does this help ?

https://docs.chainlit.io/deploy/api#use-chainlit-apis-in-your-endpoint