External HD slow to mount every time I boot up Mac after update to Sonoma

Sonoma introduced a number of low-level changes, especially around disk management and background tasks like Spotlight indexing, privacy permissions, and new security checks. What you’re seeing is often related to the OS re-verifying drives or re-indexing them on each boot, particularly if the drive isn’t formatted APFS or macOS Extended (Journaled), or if it contains large numbers of files.

It would also help if you can let us know the make & model of your external drive and how it is connected to your Mac mini.

Regardless, here are a few things you can check/try to, hopefully, "speed things up:"

• Since you didn't mention it in your post, I suggest starting by verifying your external drive's format. You can do this easily with the Disk Utility app. If it’s ExFAT, NTFS, or another non-native format, that can cause delays on mount.
• If Spotlight is reindexing the drive every boot, that can cause delays. Go to System Settings > Siri & Spotlight > Spotlight Privacy and add your external drive to the privacy list to stop it from indexing.
• Check System Settings > General > Login Items, and see if anything is trying to access the drive at boot. Note: If you use any third-party automount tools or backup software (e.g., Carbon Copy Cloner, Time Machine), disable them temporarily to test.
• Depending on whether your mini is an Intel or Silicon model (I'm assuming that it is the former,) try resetting its NVRAM and SMC. NVRAM stores info about connected drives; a reset might help.
  • Reset the SMC on your Mac - Apple Support
  • Reset NVRAM on your Mac - Apple Support
• Boot into Safe Mode. Let the system load fully, then reboot normally. This can clean up caches and fix odd delays.
• Finally, use Disk Utility's First Aid to check the drive's "health."

You're very welcome! I'm glad you got it worked out.