'a' is an article preceding a noun beginning with a consonant and 'an' a vowel.
What about a noun beginning with a vowel (egg) is preceded by an adjective beginning with a consonant (big).
Is it 'a big egg' or 'an big egg'?
It simply depends on the next phoneme after the article. If it sounds like a vowel, then use an. The letter doesn't even have to be a vowel: an hour
So, the answer to your question is, of course, "a big egg."
Additionally, use "an" only if the next "sound" is a vowel, like this,
a + egg => an egg
('e' is a vowel "sound")
a + big egg => a big egg
('b' is not a vowel "sound")
a + hour => an hour
('h' is not a vowel "sound", but it's not pronounced. It's silent. The letters that are pronounced are "ou". They are vowel "sounds", so "an" is used.
a + long hour => a long hour
('l' is not a vowel "sound")