It sounds as if it hasn't been backfiring (exploding in the exhaust system) but spitting (exploding in the intake system). There are three common causes of that: grossly wrong ignition timing, lean mixture, and leaky inlet valve. In the case of a B&S engine an ignition timing error seems unlikely. If it were lean mixture, it would be spitting as much while it was running as it did when you were starting or stopping it. Hence it sounds like it needs attention to its valves.
