I'm not a doctor, blah blah blah, but...
Over the past several years of lifting weights, playing with strongman training and rock climbing, I've had numerous injuries including (but not limited to) a fractured thumb, popped finger tendons, severe shoulder trauma (can't remember exactly, but only narrowly avoided surgery), rotator cuff issues with both shoulders and more recently a dislocated ankle.
Throughout all of this, I've never stopped training, I've just adapted what I do (upper body broken = train lower body and vice versa).
Though I can't find a specific article, Dan John talks, in some of his podcasts and interviews, about the time that he fractured his wrist (or something very similar) and carried on training everything that he could, including his non-injured side. He found that training the non-injured side actually had a surprising amount of carry over to the injured side.
I can't comment on your particular injury, but when I injured my right shoulder (by being an idiot and not letting go when my foot slipped on a bouldering problem) I started deadlifting every day using a program I read about in Easy Strength. Once I'd got a bit bored with that, I made a routine based around some machine work, lunges and split squats (dumbbells held at my side, didn't irritate my shoulder), dumbbell bench and overhead press with my left arm, lots of core work, deadlift variations (snatch grip pulled too much, so it was mainly conventional, sumo and Romanian), single arm pulldowns and cable rows, and once the injury had healed sufficiently, daily Turkish get-ups.
I've never liked doctors for physical injuries like this. The only advice I've ever been given was "rest it", or at best, a print out of some general exercises that they seem to hand out to everyone who goes in with that particular injury (there was a pile for ankles, one for backs, one for knees, etc, etc).
Go see a decent sports physiotherapist, or a personal trainer who has experience with sports injuries.