Looks like specific settings in BIOS cause hardbrick (no video, no beep, no-POST, everything else looks "fine") to 2 of my motherboards (Dell 7010 DT and HP 8300 SFF)
Clearing CMOS/removing battery + discharging capacitors with power button in both cases doesn't have any effect.

What tools do I need to try an external reflash without desoldering the chip?

[GUIDE] The Beginners Guide to Using a CH341A SPI Programmer/Flasher (With Pictures!) follow this guide for general instructions.

@Majaque - You need CH341A and SOIC8 test clip and cables or SOIC16 (if any BIOS chip 16 pins), both around $2.50 each on ebay, if you need links let me know.
Here is two other guides as well - [GUIDE] Flash BIOS with CH341A programmer
[Guide] Using CH341A-based programmer to flash SPI EEPROM