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You can make vegan cheeses with nut milk. Here is a video for vegan blue cheese[1]. The basic idea is nut milk and the culture for the blue cheese. [1] https://m.youtube.com/watch?v=cxMAl_LiSUU&pp=ygUZR29ydW1ldCB... Channel is “Gourmet Vegetarian Kitchen”
Sorry, I wanted to know about the egg cheese
Egg cheese recipes tend to have milk in them any way, so you would likely not avoid needing a milk substitute.