I never, ever need to freeze mine because I use what is known as the 'split method' of milk soaping, which allows me to avoid having to dissolve my lye in my milk. (I suck at the 'frozen/slushy' method and have no patience for it.)
With the 'split method' I split the total liquid amount for my batch into 2 halves (half water/half goat milk or coconut milk). I mix the lye with 50% water, and I mix the rest of the 50% liquid as fresh goat milk*, or canned coconut milk in with my oils either before or just after adding the lye/water solution to them (*I use the refrigerated, fresh kind of goat milk sold in cartons in the cold case at the grocery store). This will give you a 50% milk soap. For a 100% milk soap using the split method, I just add enough goat milk powder or coconut powder to the fresh milk to bring the concentration level up to what would equal a 100% milk concentration for the total liquid amount of my batch, and then I stickblend that into the oils.
I gel all my milk soaps, and unless I use a discoloring FO, they all come out a creamy off-white color (both for GM and CM). If I add a little TD, they come out quite white.
IrishLass