I used to get perms twice a year from the time I was 13 until I was well into my 20's ~ I haven't smelled anything that bad while making soap yet . . . Yet! Thank goodness! That would be like a traumatic flashback!
Is it supposed to happen with all milk soaps? Or maybe it's more likely with milk soap recipes that have certain other ingredients as well? Because come to think of it, when I made my oatmilk soap, I made my oatmilk with almond milk and soaked it in the fridge for several days, then blended the oats into the milk for more of a milky slurry, added a little more almond milk and froze it, then made my lye solution, then then made the soap, and that soap got real hot and did have a bit of an odor when I poured it into the mold, but I slid it into the fridge real quick. I never noticed any strong fumes in the fridge though and it didn't scorch so it ended well, thank goodness!