Thermocouple usually employs a dedicated IC to provide certain required functions like reference, compensation and amplification. An usual multimeter might have an error like 1-2-3% +/- ~-1-2 digit(s) and that means error addition to the measurement, loosing a lot of precision, and 1mV/deg is quite a low scale for a common multimeter. Personally as a matter of opinion i would rather go for a multimeter which is specified to read temperatures, or a dedicated tool like your link, or even an IR thermometer - well, its just my opinion.
Edit: If you have thermocouples physically, and if you use a firmware which supports chips like max6675 you could make a small board with that chip and have the temp reading over spi. If the firmware does not support max6675 expressly, you can use some other chip which outputs an analog signal, one chip is ad595 and i think there is some other its successor, and with the latter one you can use thermocouples directly to the analog pin, but from experience that may include some crosstalk with other wires (i had like 1-2-3 deg of variation per reading accounted to the noise), whereas with max6675 talking over spi such noise should not be an issue.
Edit: If you have thermocouples physically, and if you use a firmware which supports chips like max6675 you could make a small board with that chip and have the temp reading over spi. If the firmware does not support max6675 expressly, you can use some other chip which outputs an analog signal, one chip is ad595 and i think there is some other its successor, and with the latter one you can use thermocouples directly to the analog pin, but from experience that may include some crosstalk with other wires (i had like 1-2-3 deg of variation per reading accounted to the noise), whereas with max6675 talking over spi such noise should not be an issue.
