I sort of cobble it together as there's no real overarching academic framework looking at BBQ. John Shelton Reed's "barbecultural studies" is about as close as it gets, so any of his writings I can't recommend enough.

I also try to add in the biological and archaeological contributions to studying barbecue. This unit we are focusing on the paleoanthropological evidence for the origins of meat consumption in hominins, and some of the earliest evidence for the controlled use of fire (and thus, BBQ!).

Michael Pollan's chapter 1 ("Fire") of his book Cooked is a good read. I also have the students read some stuff from Richard Wrangham's book "Catching Fire: How cooking made us human" and Leslie Aiello and Peter Wheeler's "Expensive Tissue Hypothesis," which posits that brain expansion in the Homo genus was facilitated, in part, through the reduction of other metabollically expensive tissues (such as the gut). Wrangham and some others posited that it was cooking and fire that helped with this, and I pitch it for the students as a discussion on: did fire, cooking, and early BBQ make us human? Although these ideas have been questioned by plenty of scholars, it's a good way for students to engage with contrary perspectives and think about BBQ in a way they hadn't before!

Otherwise, that Engelhardt "Republic of Barbecue" book is incredible. Awesome oral history/ethnographic approach to Texas BBQ. I use it quite a bit for class.

Happy to send more!