Speaking only for myself, I split my liquid amount when making milk soap because I passionately hate mixing my lye with anything but water. All the times when I've dissolved it directly into my milk, it was problematic for me....if it wasn't the milk getting scorched, then it was the fat in the milk saponifying before I could add it to my oils, etc...
Things just proceed much better for me when I dissolve the lye in an equal amount of water and add the milk to my oils. My milk soaps come out beautiful with a nice creamy, off-white coloring this way instead of tan.
That's not to say that one cannot get a nice off-white milk soap doing it via the direct way with frozen milk, but it takes way more time and patience than I'm willing to invest.
IrishLass