Years ago I was approached by an Application Engineer with "I have 5 degree Centigrade error in this circuit, using a type-K thermocouple. What to do?"
Over the next few days, we uncovered several principles for such circuits and such PCBs.
place the local (cold junction) temperature sensor in between the 2 thick pins of the special type-K socket
create a thermal short around that type-K socket, so both pins have the same temperature
have NO HEAT dissipation near the type-K connector, so thermal flux GRADIENTS can be taken to near_zero
do not use a high_power MCU to perform the math.
do create a thermal open between the type-K connector(and cold junction sensor) and the MCU and other heat dissipating components (such as USB interface or LDOs)
be aware of the HEAT from you own forehead, flowing into the copper of the PCB and causing thermal gradients.
How to do this?
Use 4 layers of PCB; around the type-K connector you must thermally connect 3 of the layers with vias every 1cm, so those vias and layers as ensemble operate as near_isothermal region, ensuring the 2 type-K pins are at same temperature and that the cold_junction sensor is indeed sensing only the type-K (thick copper lugs 2mm thick by 6mm wide) temperature.
Why does this work? Copper foil of the standard thickness is 70 degree C thermal gradient per square of foil per watt of heat flow.
A typical via between layers is about 1:1 aspect ratio: 1/48" diameter, 1/16th" depth if 2_layer PCB. Such a via in 4 layer (1/16 total stack height) has 3x shorter aspect ratio (1/48 diameter, 1/48 depth ==> 1/16 periphery, 1/48 depth), thus have 70/3 = 23 degree C thermal gradient per watt of heat flow.
If you have 20 such vias spread out over a 2" by 2" region where the type-K connector (and the cold junction sensor) are placed, and your heat comes from only your FACE, assuming the sensor is installed symmetrically between the 2 pins of type-K), and you have THREE layers (the 4rth for the cold junction sensor wiring), and you have ONE WATT heat intrusion from your face/forehead,
then the thermal error is
23 degrees / ( 20 vias * 3 un_slit copper foil planes 1.4 mils thick)
23/60 = 0.4 degree error at most
What success did the Application Engineer have? his 5 degree error immediately dropped to less than 1 degree, and since then he has created his own business line of thermocouple PCBs.