Start out measuring enough goat's milk to equal the amount of water in your recipe. Put it in ice cube trays or a ziploc bag and freeze it to at least a slushy consistency. Break up the frozen milk and put it in your pitcher. Stir until it's lightly slushy and then start adding your lye, bit by bit, very very slowly.
Wait several minutes between adding more lye. Stir a bit. Wait a bit.
You don't want the temperature to get much over 90 degrees. Keep adding tiny bits of lye to the mixture until it's all mixed. For a normal batch of lye-milk, this will take me 10-15 minutes.