In an Egg, or any cooker where the food is exposed to burning coals, most of the heat energy is in the form of IR radiation, not hot gases. As is the case with all such radiation, the intensity falls off by the inverse square. Raising the food just a few inches can prevent burning, but still promote crisping on the side facing the coals. This makes it a good position for doing smaller items, burgers, chicken parts, etc. that can cook quickly w. a few turns, and which most people like nicely browned on the outside.
The turkey stock should be ok. Looks like it is very concentrated. The dilution should help the turkey flavor from overpowering the pork tenderloin. Lots of good savory veg flavor in there that will pair w. the stuffing.
Its curious. People cook pork all the time. But the market only has pre-made beef, poultry, veg, and fish stock. Seems weird to me. Never seen even red-eye gravy on a shelf.
Good luck, have fun.
When you have time, learn to make stock. Homemade is ++ better than store bought. Learn to make a roux. Sooner or later someone will call you Escoffier.
Not sure a thermopen would be much help in this situation. The patties might be thin enough that the probe tip either reads the cool center or the cooked exterior depending on a very small distance.
I found some reviews of the Blumenthal pre-made patties. They all cooked the burger fast in a hot searing pan. with frequent turns. The pans would transfer a lot of heat really quickly. This would produce a well seared surface, and I would suppose a still slightly pink interior. They were also coated in some sort of fat before the sear. That would help a Maillard crust form.
One of the classic low tech methods of knowing when pan seared foods are ready is that they stop sizzling, and lift pretty cleanly from the pan. Cooking in an Egg won't be the same unless you are using a CI griddle.
I'd go with a hotter dome temp, 450F - 475, cooking raised, direct. I think that would give a similar effect as using a pan w. oil near its smoke point, which would be around 400F. There will be flare-ups, and one will need to work quickly. It would be even better to cook at the lower position, but flipping the burgers is harder, and there would be lots of flame.
What I would look for is just how well browned the exterior is. I would hope that the patties are thick enough that the outside will be done while the inside is still pink. I could only find 1 pic of a Blumenthal style burger that had been bitten, and it seemed fairly reddish.
Old fashioned test, after a couple of flips, make a slice in the edge of one to see what going on inside.
Check out the Bernzomatic JT850. Can use either Map Pro gas, for super fast starts, or propane. The long barrel keeps the nozzle about 3' away, but overall, its light and small enough that its a lot like carrying a cane.
A good discussion of food safety temperatures would take pages. Do read around, there's lots on the topic.
In short, "4 hours between 40F & 140F" was a very broad statement designed to insure food safety in restaurants. Over the past few years it has been criticized, and I have come across a few restatements.
The problem is that the growth and death rate of pathogens depends on temperature as well as time. Very roughly speaking, all pathogens will be found growing as their temperature approaches 40F. The growth rate doubles roughly every 10 warmer it gets, until around 100F. At that point the growth rate slacks off. Around 126 the pathogen population stop increasing. At 140, the organisms are being destroyed rather quickly. Around 185, they are destroyed almost instantaneously.
The rule supposes that the food has been properly handled until it reaches the kitchen, and has little or no contamination. Once food preparation starts, it is very likely to become contaminated. Humans carry staph, for instance. Botulinum may reside on the outside of garlic. Etc.
Given slow growth at lower temperature, working in a 50F kitchen would give a longer period of safety. But "if you can't handle the heat, stay out of the kitchen" is the normal situation. Pathogens couldn't be happier. One rule revision I came across (if I recall correctly) was that 2 hours was the safety limit at 90F.
Note that even tho' the pathogens may be dead at140, the toxins they produced may still be present. Staph is particularly nasty, because some strains produce poisons that do not break down until over 250F.
And another "gotcha" is that once food is cooked, it is less resistant to pathogen growth. The broad safety rule is that cooked food can only be in the 40 - 140 danger zone for 2 hours.
If you look at the rules for how hot a food should get to be safe to eat, you'll notice than things like ground meats aren't safe till 185. The notion is that the process of grinding would spread pathogens thru the whole mass. Some larger pieces meats may be served after reaching 125, because only the surface was contaminated, and even an oven at 200 would quickly wipe out any pathogen on the surface.
Many of the foods cooked in the Egg are safer than average. The salt and sugar in BBQ rubs, as well as some of the herbs and spices, destroy pathogens. The smoke destroys pathogens. Vinegar based mops and sauces kill pathogens.
I do follow the adage "when in doubt, throw it out," but one of the most common Egging problems is when the fire goes out during an overnighter and the food is discovered sitting at 120 or lower, and one wonders if the food is still safe. 1st, check the food temp. If still above 140, no prob. I did a couple test a few years ago, I shut down an Egg with a platesetter in it when the dome temp had already declined to 180 because the fuel was running out. The exterior temps were in the high 30s to low 40s. It took longer that 1 hour for the dome temp to drop to 120.
There area of debate remains on how safe the meat is sitting in the Egg. Even at low temperatures, the surface of the food exceeded 180 quite rapidly. The interior of the Egg is not hospitable to pathogen growth. Too dry, too much formaldehyde and creosote residue. The vents don't allow much air flow. While in cool temperatures, fecal dust can carry pathogens (got a neighbor that doesn't clean up after Fido?), hotter temperatures and sunlight clean the dust. My opinion, and that is all it is, is that meats that have made it thru long cooks are pretty safe sitting inside a cooled eggs, covered with their crust of rub and smoke.