MacBook Pro turn off when unplugged an external monitor

Since there is a shutdown / restart, Checking the log for the shutdown cause should help to troubleshoot.

Below command in terminal should help to get the reason.

log show --predicate 'eventMessage contains "Previous shutdown cause"' --last 24h

"24h" is the duration, adjust to your approximate time when this occurred.

Also, check Activity Monitor to see the max CPU/RAM usage when you disconnect/unplug HDMI cable.

Have you checked if this occurs in safe mode too?

Start up your Mac in safe mode - Apple Support

See: https://eclecticlight.co/2017/02/28/mac-shutdown-and-sleep-cause-codes/

some of the known codes for the shutdown cause.

" -128 unknown, possibly battery is at the end of its life, but can also occur when the SMC initiates an automatic restart following a kernel panic."

Since you mentioned you brought new battery, did you had this problem with original battery that your MacBookPro came with?

Also, what is the MBP model?

Suggest you to Use Apple Diagnostics to test your Mac - Apple Support and visit/get it checked with Apple Support/Service. {Mac Repair & Service - Apple Support}