The first step is to find a suitable peltier module to use.  Let's try to scope the requirements.  To measure and report temperature should take well under 100mW.  We don't have any significant size restrictions.  I don't have a big prototype budget and one of the goals is to find a cost effective solution, so price and availability are important.  

A little searching online makes it pretty clear that thermo-electric-coolers (TEC) are more common than thermo-electric generators (TEG) and so the price and selection of TEC is significantly better than TEG.  There are many examples of people using the TEC devices for generation, even though their generation capabilities are not usually characterized very well.  Most of the people trying to use these for power generation are trying to extract much more power than I need, so I feel pretty comfortable that the TEC1-12706, which seems to be a popular low price module, will be enough for this project.  

The maximum working temperature for this module is 150C.  This is significantly less than the maximum temperature inside the BBQ, so we need to be careful to shield it from the heat.  There are higher temp devices, but the cost is significantly higher.  We may need to revisit this selection, but it should be good enough for the initial prototypes. 

I'll add the TEC1-12706 to the component list.