Add mount_path support to FastMCP for proper SSE endpoint handling

This commit adds a mount_path attribute to FastMCP settings to support
properly constructing SSE endpoints when servers are mounted under
sub-paths. The _normalize_path helper method combines mount paths with
endpoints to ensure correct session_uri generation.

This fixes issues when running multiple FastMCP servers under different
path prefixes in a single Starlette application.

Author: tim-watcha

src/mcp/server/fastmcp/server.py

@@ -72,6 +72,7 @@ class Settings(BaseSettings, Generic[LifespanResultT]):
     # HTTP settings
     host: str = "0.0.0.0"
     port: int = 8000
+    mount_path: str = "/"  # Mount path (e.g. "/github", defaults to root path)
     sse_path: str = "/sse"
     message_path: str = "/messages/"
 
@@ -477,9 +478,37 @@ async def run_sse_async(self) -> None:
         server = uvicorn.Server(config)
         await server.serve()
 
+    def _normalize_path(self, mount_path: str, endpoint: str) -> str:
+        """
+        Combine mount path and endpoint to return a normalized path.
+        
+        Args:
+            mount_path: The mount path (e.g. "/github" or "/")
+            endpoint: The endpoint path (e.g. "/messages/")
+            
+        Returns:
+            Normalized path (e.g. "/github/messages/")
+        """
+        # Special case: root path
+        if mount_path == "/":
+            return endpoint
+        
+        # Remove trailing slash from mount path
+        if mount_path.endswith("/"):
+            mount_path = mount_path[:-1]
+        
+        # Ensure endpoint starts with slash
+        if not endpoint.startswith("/"):
+            endpoint = "/" + endpoint
+        
+        # Combine paths
+        return mount_path + endpoint
+    
     def sse_app(self) -> Starlette:
         """Return an instance of the SSE server app."""
-        sse = SseServerTransport(self.settings.message_path)
+        # Create normalized endpoint considering the mount path
+        normalized_endpoint = self._normalize_path(self.settings.mount_path, self.settings.message_path)
+        sse = SseServerTransport(normalized_endpoint)
 
         async def handle_sse(request: Request) -> None:
             async with sse.connect_sse(