The correct answer is it depends on the thermometer you are using.
Here is an example, where someone took a cheap infrared thermometer from harbor freight ($25) and used it to control the temperature of a hot plate so he could do surface mount circuitry.
To know if your meat thermometer will work you will have to get into the technical details of how it works, and how you can connect to it. But really, there is no need in banging your head against the wall, someone has already done that. Just Google "Arduino Meat Thermometer". People have made control circuits for smokers, for BBQs, some even have a web interface to monitor the temperature.
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I have not worked with thermopiles before.
Essentially, for the best performance with Arduino (and Galileo), you would like to get a voltage between 0-5V or 0-3.3V. I don't think the thermopile outputs those voltages. However, it does output some voltage. That voltage can be amplified by an OpAmp, so that it is in a nice measureable range for the analog inputs of the Galileo.
Here is someone who wrote an instructable with a different thermopile, but the circuit should be similar (if not the exact same) Thermopile Sensor.