The temp sensor needs to be economical and work at high temperatures.  A BBQ is too hot for thermistors so thermocouples are probably the next most cost effective.  There are chips that can digitize thermocouples like the LTC2485 and the MAX31855.  The price seems similar, but the MAX31855 is tuned for the specific type of thermocouple so there will be less software processing.  The K type covers the temperatures of interest so I'll add the MAX31855KASA+ to the component list.