It's from The Last of Us video game:
A boiled egg in the morning is really hard to beat.
I don't get it.
Here are more puns if somebody is interested. I managed to understand the rest of them though.
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One of the senses for the verb beat is, per Wiktionary:
To mix food in a rapid fashion. Compare whip.
"Beat the eggs and whip the cream."
Thus, you can only "beat an egg" if it is in the raw condition.
Secondly, the expression "hard to beat" has the idiomatic meaning (Macmillan):
to be very good, or to be better than other things of its kind
"It’s an expensive hotel, but that kind of service is hard to beat."
So, there are two layers of meaning here:
It is hard to turn a boiled egg into a mix by beating it (I would say, almost impossible). Since it is hard to do this in any part of the day, naturally it is hard to do this in the morning.
It is great to eat a boiled egg in the morning: you can eat something else, but that would most likely be less delicious than a boiled egg.
Meaning 1: Boiled eggs for Breakfast: YUMMO!!!! It's "hard to beat that", i.e. it's hard to better that as an experience. Frank McCourt in "Angela's Ashes", describing his family's childhood poverty in Limerick Ireland, dreamt of being sooo wealthy that his little brother Alfie could have a boiled egg everyday if he wanted one and how wonderful such wealth would be.
Meaning 2: To fold an egg, as in cooking (i.e. stir vigorously either to mix, or in the case of eggwhites, to beget chemical change that gets them ready for making meringue or Pavlova). Eggwhite only folds when the egg is raw. You can try it on cooked eggwhites, but only if you have a really powerful blender!
If you break a boiled egg and mix the white with the yolk (yellow) no matter how you do it, I would be much surprised to see it that way in the morning for breakfast unless you want to put it into an egg salad.
When you make an egg salad, you use the verb to "chop" (cut something into pieces with repeated sharp blows). The purpose is not to mix the white with the yolk as to "beat" does, but to break the boiled egg into as many pieces as possible.
The below is an excerpt from Timaru Herald in 1898. The expression "hard-boiled egg" has been used for a long time to refer to a tough guy (if you are a tough guy, it is not easy to beat you) or something that is second to none (best in quality).
2 May 1898, Timaru (NZ) Herald, pg. 4 ad: WHY is BENJAMIN GUM like a Hard-boiled Egg?—Because it is HARD TO BEAT!
Your sentence has a double meaning as well explained by the other answer.
A boiled egg is the best food you can have in the morning.
Nobody can beat a boiled egg because nobody can. If you do it, you would be considered "crazy".
I don't think it is a matter of whether it is possible to beat a boiled egg or not. We don't beat a boiled egg because it is not customary. And if something is refered to as a "hard-boiled egg", it has the best quality and it is almost impossible to beat its quality. The two meanings are closely related.