I know a self-defense fighter who broke both fists fighting. What self-defense techniques would everyone recommend to a fighter forced to fight with two broken fists?
Before getting to your question, I'd recommend an art and practice routine that teaches how to form a proper fist and condition it so it won't break easily during a fight. IMHO, practice with heavy gloves tends to teach people bad habits, such that they easily injure their hands in a fight. Conditioning using some moderately hard target like a makiwara is a good idea.
It's not clear whether you mean someone's injured their fists, and then wants to find a martial arts training routine in case they have to fight one day, or they unexpectedly injure their fists during or immediately before a fight and want to be able to continue regardless. I'll assume the latter.
If the fists are injured, the more alternative tools you can still use well the better.
ARMS: you may or may not be able to use the palm to strike, but you should be able to use forearms and elbows.
Arts like Muay Thai and karate train to throw elbows.
Many arts use palms - particularly Chinese systems.
A few arts use forearms - some kung fu styles and I suspect some Indian martial arts, but I've not trained in any myself. That said, karate (and derivatives like taekwondo) have haito uchi ("ridge hand" or "reverse knife hand" strikes), and the mechanics can be used for forearm strikes: you just need to be 10cm closer.
LEGS: clearly two broken fists is a much bigger problem for a boxer than a kick-boxer: anyone trained in kicking and kneeing can rely more heavily on those as needed.
HEAD: A very few arts actively train headbutting - Myanmar's Lethwei is very strong in this regard, and is otherwise generally similar to Muay Thai.
THROWS/GRAPPLING/JOINT-LOCKING ETC: There are still take downs and ground fighting moves that don't require use of the hands, but it'd probably be difficult to spontaneously fight when your training habits involved grabs and holds if you'd no prior practice restricting yourself to the subset you could manage with seriously injured hands.