How does said software get "flashed" onto the hardware?

The UEFI firmware is programmed onto a chip before assembly of your mother's motherboard. Although it can be updated afterwards by the end user.

The UEFI applications that you can build with Lil UEFI can be shoved into a special FAT32 partition on your disk and named EFI/BOOTX64.EFI[1], the firmware finds this file and will start executing it.

[1]: It can be named something else so long as you tell the firmware about it in advance. I think that you can also load them via netboot.

In a production environment, the eeprom chips will get flashed before they're placed on the board and soldered.

In a DIY environment, https://www.flashrom.org/ or TL866 or something.