How to call repository_dispatch via Project Integration Webhook?

How do I call a github webhook with… what ever the hell this is?!?

Githubs repository dispatch webhooks need some required payload fields, or they will return 404 with a friendly "documentation_url": "https://docs.github.com/rest/repos/repos#create-a-repository-dispatch-event" payload:

It also requires a different payload type than just json.

A typical request would look like this:

curl -L \
  -X POST \
  -H "Accept: application/vnd.github+json" \
  -H "Authorization: Bearer <YOUR-TOKEN>" \
  -H "X-GitHub-Api-Version: 2022-11-28" \
  https://api.github.com/repos/OWNER/REPO/dispatches \
  -d '{"event_type":"unity webhook", action":"webhook","branch":"staging"}'

(+ a client payload that tells you which build number, for example)