Since, all I make is goat milk soap. I can give you some pointers with working with milk in soap.
First, I freeze my milk SOLID....right now I have about 120 pounds of milk frozen in the deep freeze. I weigh out the amount that I need for a batch...mine is 1 pound. So, I put the milk into gallon ziplock bags and freeze in about 3 inch wide and inch tall bags...I press the air out and roll the rest of the bag over twice.
After I weigh out all my oils and set them aside, I grab one of those bags and put into hot water. Measure out my lye, and check to see if the "tube" of milk will break. Once, I get a break I dump the milk out into my lye bowl and add the lye. I would not say that I add the lye slowly...as it just takes me 20 minute from weighting out the oils till I am washing up the mess. But, I add at a steady flow and stir the whole time I am adding lye. Now, I do have my bowl in the sink with just cold tap water around it. It will turn a yellow pudding looking thing. Most of the time I have the lye all mixed in before the milk is even done thawing out. That is my goal at least.
I use no water in my soap...as my customers want only 100% goat milk.
Then once, I get the soap batter mixed I just pour it into my molds. I use lined wooden molds...mine are 36 inches long and 3.5 inches wide. I dont put anything on top of them or insulate at all. As most of the time I am running out the door to my "real" job. Then at night, I just pull out the soap logs and let set for about 2 to 3 days still in the paper. As I have found if I let them sit that long they are easier to cut.
Hope this helps.
ken