A chicken and onion with egg cooked with special sauce served top of a bowl of rice.

Oyako means "parent 親[oya] (chicken) and child 子[ko] (egg)" and domburi 丼 means "rice bowl dishes".

Oyako-dombri is popular among Japanese people.

We also have "Tanin-domburi".
Tanin 他人 means "anyone else, others, stranger, etc.".
So "Tanin-domburi" is a bowl dish with beaf and egg..

If you would like to taste it, I recommend you Sagami chains in and around Nagoya (reasonable and tasty).
The name of "oyako-domburi" in that restaurant is "Nagoya Coachin-don"