On Oct 30, 2022, publishing to a custom domain on FlutterFlow was announced via a newsletter:

To start you can publish one custom domain per Pro or Teams account, additional options coming soon!

Looking at the Pricing, they encourage you to pay annually (25% discount). Also, the Pro and Teams tiers are the most expensive, at about $50/mo. That's a bit much just for that one feature that I want to use. But oh well. They do save me a lot of time, and it's worth that much.

The doc hasn't been updated yet.

Going into App Settings, Web Publishing, I input my custom domain.

The UI then tells me to set up two records on my registrar. An A record and a CNAME record. Once I update my domain, I clicked "Connect" on FlutterFlow, and it worked. It also showed me that I get one free connected domain:

I really hope further custom domains don't cost too much.

Cheaper workaround

One other way to publish to a custom domain would be to get the Standard subscription ($25/mo), download the code, compile it to Web myself and deploy to Firebase Hosting. However, the Standard subscription does not have Git integration. Only Code download. That means it might be a bit of a manual task to download the code, then compile and deploy.