I have a 2018 Mac mini (Intel, of course) with two monitors. One is a 4k Dell plugged in via HDMI cable to the HDMI port on the mini. The other is an older Dell plugged in via a USB-C-to-HDMI adapter and an HDMI-DVI cable. Ever since upgrading to Big Sur, I've had problems getting both monitors to work. With both monitors plugged in, the 4k monitor stays black on boot and after I log in. I was able to workaround this by unplugging the USB-C-to-HDMI adapter, booting, logging in, then plugging the adapter back in. That got both monitors working, but was a real pain.
I think I finally have a more reasonable workaround, though. Yesterday, I worked with Apple support on this and they had me delete my windowserver preferences files, located at:
/Library/Preferences/com.apple.windowserver.plist
/Users/[me]/Preferences/ByHost/com.apple.windowserver.*.plist
Then, they had me shutdown and reset NVRAM (holding Option-Command-P-R for two chimes). This fixed the problem temporarily, but if I restart or let the computer sleep I will lose the 4k monitor again.
Even without the preferences files, restarting seems to always leave me with one monitor. However, shutting down and cold booting seems to result in both monitors working (after I log in, anyway).
So, my recommendations are:
1. Shutdown and power on again instead of restarting
2. Set your Energy Saver prefs so the computer doesn't go to sleep. (Display sleep seems to be OK.)
3. Delete the preferences files I listed above if necessary.
4. Reset your NVRAM if necessary.