What does 도른자 mean?
I think it's a slang used by young people, I tried to look up the dictionary but it's not there.

Question

Answer

Yes. I think It's slang. 
The word '도른자(crazy people)' is a new word.

돌은 + 자 = 돌은 자 -> 도른 자
 
'돌은' is turn or go nuts adjectives.
'자' is persion noun.

'돌은 자' and '도른 자'
Though phonetically alike,they are written with differentideographs.

* Bats in the belfry
This means that someone is very eccentric or even crazy.

Answer

Maybe you heard 노른자 not 도른자.
노른자 means egg yolk. In many cases, 노른자 means the essence, or highest-priced land of an area, heart, kernel, etc.