Option to mount SSE server to existing ASGI server (#312)

* Option to mount SSE server to existing ASGI server

Fixes #311

Add option to mount SSE server to an existing ASGI server.

* Add a new method `sse_app` in `src/mcp/server/fastmcp/server.py` to return an instance of the SSE server app.
* Update the `run_sse_async` method in `src/mcp/server/fastmcp/server.py` to use the new `sse_app` method.
* Update the documentation in `README.md` to include instructions on how to mount the SSE server to an existing ASGI server.
* Fix the example in `README.md` to use `app.mount('/', mcp.sse_app())` instead.

---

For more details, open the [Copilot Workspace session](https://copilot-workspace.githubnext.com/modelcontextprotocol/python-sdk/issues/311?shareId=XXXX-XXXX-XXXX-XXXX).

* Add `sse_app` method and update `run_sse_async` method in `server.py`

* Add `sse_app` method to return an instance of the SSE server app
* Update `run_sse_async` method to use the new `sse_app` method

Update `README.md` to include instructions for mounting SSE server

* Add section on mounting the SSE server to an existing ASGI server

* fix: Move import statements to the top of the file/

* docs: Update README to reflect changes in mounting SSE server with Starlette

* docs: Formatting of SSE server mounting example in README

README.md

@@ -31,6 +31,7 @@
   - [Development Mode](#development-mode)
   - [Claude Desktop Integration](#claude-desktop-integration)
   - [Direct Execution](#direct-execution)
+  - [Mounting to an Existing ASGI Server](#mounting-to-an-existing-asgi-server)
 - [Examples](#examples)
   - [Echo Server](#echo-server)
   - [SQLite Explorer](#sqlite-explorer)
@@ -346,6 +347,31 @@ python server.py
 mcp run server.py
 ```
 
+### Mounting to an Existing ASGI Server
+
+You can mount the SSE server to an existing ASGI server using the `sse_app` method. This allows you to integrate the SSE server with other ASGI applications.
+
+```python
+from starlette.applications import Starlette
+from starlette.routes import Mount, Host
+from mcp.server.fastmcp import FastMCP
+
+
+mcp = FastMCP("My App")
+
+# Mount the SSE server to the existing ASGI server
+app = Starlette(
+    routes=[
+        Mount('/', app=mcp.sse_app()),
+    ]
+)
+
+# or dynamically mount as host
+app.router.routes.append(Host('mcp.acme.corp', app=mcp.sse_app()))
+```
+
+For more information on mounting applications in Starlette, see the [Starlette documentation](https://www.starlette.io/routing/#submounting-routes).
+
 ## Examples
 
 ### Echo Server