What should the company do to accomplish these goals?

Explanations:

Using an AWS Lambda function to deploy CloudFormation templates is not an ideal solution for efficient management and automation of deployments, especially in a multi-Region setup. Additionally, stack policies are not the correct mechanism to notify the team about template changes; stack policies are for controlling updates and deletions, not notifications.

Hosting CloudFormation templates in Amazon S3 and triggering updates with S3 events isn’t a proper solution for automating the deployment order or managing multi-region deployments. Additionally, Amazon SNS can notify the team, but the solution doesn’t address deployment efficiency or order.

CloudFormation StackSets and drift detection are useful for managing and updating stacks across multiple accounts and Regions, but they do not handle the specific order of deployments or efficiently address the needs for template change notifications. Drift detection only identifies when the actual configuration differs from the intended configuration.

Using CloudFormation nested stacks and stack sets allows for efficient, ordered deployment of templates across multiple Regions. StackSets facilitate deployment management in multiple Regions, and nested stacks enable modular deployments. Additionally, using Amazon SNS for notifications ensures that the data engineering team is alerted to changes in the templates. This solution meets all the goals efficiently.