According to the search results, there are several things you can do with the Githyanki Egg in Baldurs Gate 3:
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Give it to Lady Esther: Lady Esther is a character who requests that you steal a Githyanki Egg from the nearby creche and give it to her so she can raise a peaceful member of the Gith race. If you give her the egg, she will reward you with gold and appreciate your help.
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Sell it to the Society of Brilliance: If you decide not to give the egg to Lady Esther, you can sell it later on to a duegar from the Society of Brilliance.
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Keep it for yourself: If you keep the egg for yourself, it will stay in your inventory and Laezel, a companion character, will get mad at you for taking it. However, there is no known use for the egg if you keep it.
Its worth noting that the quest involving the Githyanki Egg is a side quest and not necessary to complete the main story of the game.
