Searching for the right temperature sensor with 0.001
1mK resolution is not such a big deal, and 1mK accuracy is not such a big deal for the electronics, at least over a narrow range (just expensive- a 0.03% reference resistor with a minimal tempco). 

The problem is calibration. Other than the triple point of water, which is a (relatively) easy source of 0.01°C +/-0.1mK it's expensive and difficult to calibrate the sensor to such tolerances.

Construction of such sensors is typically a loose coil of very pure platinum wire inside a quartz tube (sort of looks like a small version of an aquarium heater). Film type sensors have too much hysteresis. Called an SPRT (Standard Platinum Resistance Thermometer). Very slow response time and each unit will come with a calibration table for that particular sensor (perhaps suitable for framing) and an enormous invoice. It would likely have to be recalibrated at least annually. 

You could buy the equipment and do it yourself, but it seems a bit involved: 



You might want to read the NIST document that was the source of the above photo to see what's involved. 

Edit: Since the application is measuring the freezing point of milk, not (say) molten aluminum, we need only full accuracy over a very narrow temperature range- from 0°C to -0.25°C probably (but discovering that is your job). So it is entirely practical to use the triple point of water as a single calibration point at +0.01°C and you only need a range of about 250mK. So you're only looking at 0.4% of span accuracy. This should be relatively straightforward, and can be accomplished with a PRT and a calibration cell.