You could use relays to switch each thermocouple in turn to one thermocouple amplifier. This will introduce cold-end error due to the metal missmatch at the thermocouple connections. You can either choose to ignore the error (20degC or so) or it can be compensated for by having and RTD measure ambient temperature, which is then subtracted from the measured temperature.
Why not MAX6675 + analog multiplexer?
FYI, MAX6675 has been replaced by MAX31855, which has wider temperature sensing range, better resolution, and is available in versions for several different thermocouple types. Still has an SPI interface, but the data format is changed from the MAX6675, so libraries for the earlier part will not work for the MAX31855.
I made a break out board for the MAX31855 chip because I wanted to play with that chip too. I have the bare board up on lish.com with the caps inductors and protector. https://dcande.lish.com/
You can also get a couple as a Sample from Maxim when I did it I got 3 of them for free. https://shop.maximintegrated.com/storefront/searchsample.do?event=Sample&menuitem=Sample&Partnumber=MAX31855KASA%2B