Support Flat Config

[blimmer]

Is your feature request related to a problem? Please describe.

Eslint has moved to a new flat-configuration setup, which is the default in eslint@9: https://eslint.org/blog/2023/10/flat-config-rollout-plans/

eslint@8 hit end-of-life in early October 2024.

Describe the solution you'd like

We should migrate to the new flat config setup: https://eslint.org/docs/latest/use/configure/migration-guide

Describe alternatives you've considered

N/A - the legacy eslint config format will be dropped in eslint@10, so we need to migrate before then.

Additional context

eslint/eslint#18093 has lots of links to other configs that've migrated.