I've mentioned before the data read from the bus is about ten times as fast as the data that is written to EEPROM. As the readable data is extremely volatile, reading it must always have highest priority. This is what interrupts are used for, they do actions that have to be done before a specific deadline. In the TWI case the deadline for storing a byte from the bus into RAM is just before the transmission of a new byte starts.
Below we can see the timing of the interrupt routines. The yellow debug signal is pulled low during USI overflow of start condition interrupt handler. The green signal is flashed whenever a new byte is found in the buffer by the main function. A shared buffer is the way the interrupt routines are communicating with the main function.
Transferring the byte from RAM to EEPROM has a deadline much longer in the future. It is sufficient to finish the transfer before the microcontroller is shut down as that's when we lose the contents of the RAM. We can use a data pipeline where data is constantly copied from the buffer to the EEPROM, but this will be progressing a lot more slowly than the sniffing.
Below we see the timing of EEPROM writes. The green signal is pulled low while a byte is being written from the buffer to EEPROM. The yellow signal shows when bytes are received on the TWI bus.
As the oscilloscope captures show, the software architecture is very simple, with interrupt routines producing data to a buffer and the main function consuming that data by storing it in long-term memory.