Mark Brodsky knows the nasty things the haters say about him. Things like he’s a bully. An extortionist. A hedge-fund vulture.

But suggest that he’s litigious, and his eyes widen.

“If anyone calls us litigious, it’s name-calling,” says the lawyer-turned-distressed debt investor, who has spent two decades suing cash-strapped corporations and, on one occasion, an entire country over unpaid debts.

But Wall Street isn’t about sticks and stones. It’s about money.

And in a business with no shortage of sharp...