Doesn’t have to be anything too special. CPC do a good range or you could look at some of the specialist test& measurement companies like ATP (atp-instrumentation.co.uk) or Testo (testo.co.uk)
I can recommend item IN02030 from CPC- it’s an autoranging DMM with a type K input, RS232 out, and data logging software. Even with a cal cert it’s less than £50.
if you need differential inputs you could look at (say) IN00715.
Just need to select your choice of thermocouple.
Plus, for some applications (as long as the item isn’t too shiny) the non-contact laser-sighted IR devices are useful. Instant response! see IN02294 and similar.