Saying Japanese not to hard speak own or not.
Fastest way by google translate copy Japanese then click speaker icon(cell phone apps works off lines).
Not need translate.(in this case wrong answer both JP to EN/EN to JP)
Learn to natural speaking recommend this web.
http://voicetext.jp/Bellow copy and select speaker icon with JP flag then click "再生する(play)" button very eazy.
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すみません。私には深刻な卵アレルギーがあります。
大変申し訳ありませんが、どの料理に入っているか教えていただけませんか?
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Sumimasen, Watashi niwa shinkokuna TAMAGO-arerugiiga arimasu.
Taihen mo(u)siwake arimasenga, dono ryori ni haitteiruka oshiete itadake masenka?
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Second phrase means "Pls tell me which food include or not?".
If felt reading speed fast click "遅い(slow)" to smaller number from 100.(ex 80)
"google translate" speaking function mostly not wrong just not natural.
Also if menu with picture more simple phrase
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この料理に卵は使われていますか?
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Kono ryori ni TAMAGO wa tukawarete imasuka?
(Is this food include egg?)
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or
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こちらのメニューの中で 卵が含まれる料理はどれですか?
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Kochirano menyu no nakade TAMAGO ga hukumareru ryori wa dore desuka?
(Which food include egg these menu?)
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BTW, egg allegic all kinds? fishes,birds and more?
This is menu list of JP foods include eggs.
http://odevivi.com/allergy/alerugy/rinyu/a-8f2.htm(copy URL to google translate)