fix: use Mount instead of Route for SSE message handling

This change follows the recommended approach from MCP SDK (modelcontextprotocol/python-sdk#83)
for handling SSE messages using Starlette's Mount instead of Route.

- Replace Route with Mount for /messages/ endpoint
- Update SseServerTransport path to include trailing slash

Author: Sami Hamine

src/fastmcp/server.py

@@ -428,9 +428,9 @@ async def run_stdio_async(self) -> None:
     async def run_sse_async(self) -> None:
         """Run the server using SSE transport."""
         from starlette.applications import Starlette
-        from starlette.routing import Route
+        from starlette.routing import Route, Mount
 
-        sse = SseServerTransport("/messages")
+        sse = SseServerTransport("/messages/")
 
         async def handle_sse(request):
             async with sse.connect_sse(
@@ -442,14 +442,11 @@ async def handle_sse(request):
                     self._mcp_server.create_initialization_options(),
                 )
 
-        async def handle_messages(request):
-            await sse.handle_post_message(request.scope, request.receive, request._send)
-
         starlette_app = Starlette(
             debug=self.settings.debug,
             routes=[
                 Route("/sse", endpoint=handle_sse),
-                Route("/messages", endpoint=handle_messages, methods=["POST"]),
+                Mount("/messages/", app=sse.handle_post_message),
             ],
         )