How to burn program into EEPROM without flashing the Arduino ?

Other Hardware 3rd Party Boards
3

Using AVRdude, you can essentially read or write any part of the uC.
This article shows how.

Specifically for EEPROM:
avrdude -c arduino -P com4 -p ATMEGA328P -b 19200 -U eeprom:r:%temp%\backup_eeprom.hex:i

and

avrdude -c arduino -P com4 -p ATMEGA328P -b 19200 -U eeprom:w:\Users\owner\AppData\Local\Temp\backup_eeprom.hex

The first command uses ISP on com4 to create a backup file of the EEPROM contents.
The second command uses ISP on com4 to write the file "backup_eeprom" to the uC

Make changes in the script files to suit your environment.

Ray