Log when ESP is being reset

To what purpose ? What information do you want have ? When i had some issues with stability i added a page that would show me how long the ESP had been running using millis(). You could add a counter that increments every time the ESP has been started and save/load this counter to EEPROM, but eventually the cell will stop working.
Ideally you would want to have the reset cause as it is put on the UART, that is really useful information,