I've used CDK before to set up self-mutating pipelines on AWS (example https://docs.aws.amazon.com/cdk/latest/guide/cdk_pipeline.html). Is it possible to do the same thing with Pulumi?

you'd need to define the code pipeline manually

So I would have a pipeline stack on Pulumi in addition to my staging/production stacks?

I'm not familiar with codepipeline myself, but that sounds about right

Yes, CDK self mutating pipeline is just CodePipeline defined somewhere else. So you need a separate stack/project for CDP. I usually advise everyone to migrate away from AWS Code* tools unless it's absolutely necessary because they make things really complicated. The fact that it took almost a year for the CDK team to provide a CDK pipeline speaks for itself. CodePipeline being defined as code/configuration you have to maintain 2 states. 1) Pipeline itself 2) The actual infrastructure you are interested in deploying

I'm also considering migrating away. Thanks for the data point. I was able to set up a self-mutating pipeline in the end.

This may help you: https://pulumi-community.slack.com/archives/C01PF3E1B8V/p1637051540158300?thread_ts=1636984277.151800&cid=C01PF3E1B8V