My own theory is that the phenomenon results because the lye is stupid - put it with fats, and it's busily making soap.
I tend to freeze my GM in an oversized, plastic measuring cup. I take the cup out of the freezer about half an hour before I'm going to add the lye. That way, I've got a little bit of liquid to start dissolving the lye, and as I add more, more liquid thaws out.
I also keep my liquid container in an ice-water bath. Sometimes it stays cold, sometimes it heats up. I'm not sure the cold is the culprit in the pudding-solution problem. I think it's just the fact that there is a lot of lye with not much fat, so the lye kind of runs amok when all it has to work with is the fat in the goats' or coconut milk. (I don't ever see this if I'm using 100% beer, which was frozen before I combined it with the lye.) I usually mix my lye in whatever frozen non-water product I'm using after I have my base oils heated, while they are cooling. If my solution starts to thicken, I'll pull it out of the ice-water bath. I leave it right at the edge of the sink in case it starts to get too hot, so I can put it back in the ice bath. If it looks like its getting to thick, I just give it a stir or two, fairly quickly. That seems to make it thin out again.
One odd thing I've noticed is that sometimes if the GM or coconut milk is thickening, when I add it to the base oils, the two separate into layers, with the milk/lye solution sinking to the bottom. Once I get after it with the stick blender, though, the whole thing mixes together smoothly and uniformly.
I don't have any objection to the pudding solution, so it's not a problem for me to clean-up. I don't know if letting your frozen GM thaw out more would help??
Another warning, since you said this was your first GM: for a few days after you unmold and cut, your soap will smell PUNGENTLY
of ammonia. Don't panic, this is normal.
The smell dissipates after a few days. (Seem I read somewhere it's the initial reaction of the proteins in the milk with the lye.)
Hope this is helpful.