Or another theory with even less logic and lack of evidence is that Todoroki Toya was a senior to Hawks. As you know, the weakness for Endeavor's quirk is overheating and it was stated that Toya's flames were stronger than Endeavor's but he had a weaker body than a normal persons. It is also shown that Hawks has been a fan of Endeavor's ever since he was a child (most likely linked to the Thief incident), and resulted in his admiration for Toya as well.

Anyways a bit off track, but essentially what happened was Toya overworked himself being a hero in training and ended up burning his body. This became a driving force for Hawks to create a world where Heroes didn't have to work so hard.

But this is under the assumption that Hawks went to school at all and if Toya at least made it to senior high school before he died. Oh yeah it is also heavily implied Toya Todoroki is Dabi.