How do I set custom attributes on a user who signs up via amplify's Authenticator?

Question

I'm writing a landing page to handle redirects from AWS Marketplace, show a signup form, and if the user signs up associate a marketplace customer ID as a custom attribute in Cognito.

My app uses the amplify <Authenticator> component to render a sign-up/login form, but it doesn't have an obvious prop where I can either set additional user fields for Cognito or hook into a signup/login event or provide clientContext data to be passed into Cognito's post-confirmation lambda hook.

How do I set additional data on the user?