But what if the gap actually needs that much insulation? Maybe it's not a monster, just a hungry wall.

Buttering toast with a shovel is a perfect way to put it. I've patched a pan with a piece of metal flashing and thinset before, just to get through the day. Sometimes you just have to make it work.

Oh man, @river_hart18, that lentil steam smell... my kitchen walls got a little damp last time I made a big pot. Totally worth it for that broth though.