Trigger AppSync subscription notification asynchronously

You can leverage the concept of Local Resolvers for your task.

Asynchronous Lambda can invoke mutation on a None datasource, which will cause mutation result (can be as simple as an echo of all input arguments) to be forwarded to all subscription clients.

You can review How do I notify AWS AppSync subscribers of external database updates that aren't performed by client-side mutations? for more details and step-by-step implementation guidelines.

What is the best practice for having my Lambda invoke a mutation? Right now my AppSync API is authorized via Cognito User Pools.