I will offer a phrase and a single word.
He can dish it out, but he can't take it.
Someone easily criticizes other people but does not like it when other people criticize him or her: He’s mad at me for teasing him – he can dish it out, but he can’t take it!
Example: Donald Trump can dish it out but he can't take it
The connection between this and your description isn't apparent from the dictionary definition, but this article explains the connection. Here is an excerpt:
When criticized, narcissists show themselves woefully incapable of retaining any emotional poise, or receptivity. And it really doesn't much matter whether the nature of that criticism is constructive or destructive. They just don't seem to be able to take criticism, period. At the same time, these disturbed individuals demonstrate an abnormally developed capacity to criticize others (as in, "dish it out" to them).