On several occasions, I have found that adding frozen liquid, whether milk or just ice cubes, seems to lead to my lye not dissolving completely. Something that can help with this is to have a second lye-safe pitcher and pour your lye water from one pitcher to the other to check for undissolved lye, such as a lye cake stuck to the bottom.
When I make milk soap, I use canned milk (make sure it is not sweetened). You can use cow milk or goat milk. A can of milk is 14.5 ounces (I think), so I create a recipe that calls for 28-29 ounces of water. I am not sure what lye calculator you are using, but if I use The Sage I choose the highest recommended water amount. If I use soapcalc, I up the water from 38% to 40%.
So, assuming I have a recipe that calls for 28-29 ounces of water, I use half of that amount as water to dissolve my lye. While the lye is coolling, I put my can of milk in the freezer to get cold. (This may not be necessary, I did it the first time and the soap turned out great, so I still do it.) When the oils and lye are at the desired temp (I like to soap between 90-100 degrees), I pour the can of cold milk into my oils, then add my lye. I have had good success with this method. This is also how I make oatmeal, milk and honey, and I also add the honey and the oatmeal to the oils before addiing the lye water.
If you prefer a smaller batch, you can do that. I just prefer to not have part of a can of milk floating around.
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