If you are using API tokens, there is no meaningful sense of “installed”. Your client will need to know the site name (the part you are calling “your-domain”) a priori.
However, if you are using OAuth 2.0, then your original post makes an incorrect assumption (a common misunderstanding). Request URLs for OAuth 2.0 look like:
https://api.atlassian.com/ex/jira/{cloudid}/{api}
Your client obtains that cloudid by using the access token with this endpoint:
https://api.atlassian.com/oauth/token/accessible-resources
That way, the client is “installed” by means of a user doing an OAuth authorization code grant flow and then asks accessible-resources which sites it can access.