There's an old saying regarding our Airheads and their occasional problems:
When you think it's carburetors, it's probably electrical.
When you think it's electrical, it's the carbs.
Although this old wives tale may not apply in your situation, it's often true.
You said the engine stumbled a bit and then, "flat died". I believe that both carbs would have to fail in tandem. The Airheads will run on one carb -not too well, mind you- but they will limp you home. Case in point, I had a diaphragm fail on one carb and that side quit. The still functional other carb got me home.
You did not mention the model and year of your BMW. As there are some ignition differences twix model years, that info would be very helpful for a diagnosis.
Pull the float bowls and check for debris. Also, gently lift the floats, turn on the fuel and confirm flow by lowering the floats. Confirm fuel stops when you lift the floats.
If it was an ignition anomaly, start checking ground wires. Brown wires are 'earth'. Confirm battery terminals are clean and secure at both ends.
Also check to see what ignition coil you have. If it's the black and gray Bosch "Crack-O-Matic", that's one of the usual suspects. Check the simple stuff first. It's usually the simple stuff.
BTW, what shop did your work?