In the meantime, I've tried several recipes for GAPS burger buns. Most of them are glorified air, much like the puffed discs you find numerous fast food places. Air is good. The main purpose of buns is to be a delivery device, right? However, given our limitations (oxalate avoidance, minimal cheese) I wanted to figure out something that didn't involve nut flour or dairy. And I wasn't so concerned with having buns, really, I just always have a lot of egg whites!
So enough of my jabbering, check out what I've come up with. I don't think I have the technique quite perfected yet, and I've not yet tried baking these, though I've done enough similar recipes I'm pretty sure it would work well.
2 cups egg whites
2 Tbsp coconut flour
4-6 Tbsp cooked squash
1 tsp honey (opt.)
Before you even begin, turn on your griddle. You want low even heat. First fluff egg whites with electric mixer, you don't really even need peaks to form, but getting to soft peaks does mean more air at the end. Then sprinkle coconut flour over egg whites, fold in a little, then blend in rest of ingredients, folding them in gently and then beating on low speed for a minute till just combined. Cooked on low heat on a greased griddle, using a spatula to make little rounds, about 1/2 inch thick. The batter in the bowl will separate some, just refold gently as you go along (you could use more coconut flour, but I don't care for the texture). After frying, I prefer transfer to a barely greased pan to finish in the oven.
When all the buns are finished, bake at 350 for 10-20 minutes to finish insides. Or you can just leave on the griddle on low heat for a bit longer. If you bake, then they'll be warm and ready for the burgers!
Alternatively, you can line pans with parchment paper, and bake instead of frying. 300-325 degrees for 30 minutes or so depending on thickness. Parchment paper will help prevent sticking, though I have well loved stones and grease that seem to do the job well. I just prefer the more non-stick crust you get with pan frying.
You could also make these up without beating the egg whites, but that would make crepes, not puffy flat bread.
Ultimately, these do deflate and end up more like the mashed down buns you have with one of those fat burgers, but who cares? It tastes similar enough to cheap white buns to make the meal happy.