The correct answer is no article.
The word “egg” is a non-count noun, which means it does not have a plural form and is not preceded by an article. For example, you would not say “I need an eggs for this recipe.”
In this case, the speaker is not referring to a specific egg, but rather to eggs in general. They are saying that they need eggs to make the recipe, but they do not have any eggs on hand.
If the speaker were referring to a specific egg, they would use an article. For example, if they were talking about an egg that they had found in the refrigerator, they might say “I need the egg that I found in the refrigerator.”
In conclusion, the correct answer to the question “I need . . . . . . . . egg for this recipe, but we’re out.” is no article.