What is the appropriate way to prepare the Beitzah for the Seder Plate?
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The Shulchan Aruch (OC 473:4) says that the custom around him was to cook the egg in liquid. The Mishna Berura there explains that this is because the egg represents the Korban Chagigah which, unlike the Korban Pesach, could be either roasted or cooked in liquid. The Rama there notes that the custom in his city is to have the egg roasted, as was certainly a permissible way of preparing the Korban Chagigah.
The Aruch HaShulchan (:9) suggests that the custom in Ashkenaz developed out of deference to the rejected opinion of Ben Teima (Pesachim 70a) that required the Korban Chagigah to be roasted as well.
In all the sederim I have attended the egg is hard boiled. There doesn't seem to be much of a preference either way, and I would suggest each family continue in its tradition.
The Shulchan Aruch Harav says that the egg should be either cooked or fried.
Nonetheless, the footnote quotes Sefer Haminhagim that in Chabad, the egg is hard-boiled and eaten with salt-water.