Allow customization of the Starlette app (middleware, routes, etc)

[mconflitti-pbc]

Is your feature request related to a problem? Please describe.
Found that it has been useful in my fork of this repo to split out the creation of the starlette app into its own function so I can create it and customize it in my code before running it.

Describe the solution you'd like

    async def _create_sse_app(self):
        """Run the server using SSE transport."""
        from starlette.applications import Starlette
        from starlette.routing import Mount, Route

        sse = SseServerTransport("/messages/")

        async def handle_sse(request):
            async with sse.connect_sse(
                request.scope, request.receive, request._send
            ) as streams:
                await self._mcp_server.run(
                    streams[0],
                    streams[1],
                    self._mcp_server.create_initialization_options(),
                )

        starlette_app = Starlette(
            debug=self.settings.debug,
            routes=[
                Route("/sse", endpoint=handle_sse),
                Mount("/messages/", app=sse.handle_post_message),
            ],
        )
        
        return starlette_app

    async def run_sse_async(self) -> None:
        """Run the server using SSE transport."""
        starlette_app = self._create_sse_app()
        # ... rest is same

and then use this like:

mcp_server = FastMCP("example", transport="sse")

#...server set up

app = mcp_server._create_sse_app()
app.add_route(...)
app.add_middleware(...)

# run with this or the uvicorn cli
if __name__ == "__main__":
    uvicorn.run(app, host="0.0.0.0", port=8000)

Describe alternatives you've considered
none since it is currently contained in the run_sse_async function

[mconflitti-pbc]

This should also address a bug related to urllib url joining that truncates server urls paths when combininb /messages/ set in the transport object.

sse = SseServerTransport("/messages/")

For example, if joining http://localhost:8000/some/path/to/sse with /messages/, it will output: http://localhost:8000/messages

Removing the leading slash fixes this.

[panz2018]

I have developed web servers that integrate MCP SSE functionality:

• Using FastAPI: fastapi_mcp_sse
• Using Starlette: starlette_mcp_sse

These servers can be extended with custom routes while retaining full MCP SSE capabilities. Thus, it is possible to add any customization (middleware, routes, etc) in the fully developed FastAPI or Starlette servers.