You might want to take a look at Sentry:
https://github.com/getsentry/sentry
With this, you can record the errors and stacks that you usually see with DEBUG = True, aggregate them and study them more deeply. Sentry can be configured to send email to you so that you are instantly notified.
Another option that does not require a new dependency is to use AdminEmailHandler :
https://docs.djangoproject.com/en/dev/topics/logging/#django.utils.log.AdminEmailHandler
However, some information that may be required for debugging may be sensitive and should not be sent by email. That's why even the Django documents mentioned above recommend using something like Sentry.
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