You are a little short on details including such things as the vehicle year and the fuel control. I am presuming that this car is not a drive by wire car since that changes lots of stuff. Also, when you press the clutch pedal in, does the engine actually stall or are you just seeing the tach drop to zero? I am operating on the premise that when you say 0 RPM that means an engine stall.
Once up to operating temperature, does the engine idle at factory specified idle speed when the car is sitting parked and in neutral? If so, does pressing the clutch pedal while the car is parked cause the engine to stall? If it does, there is an input to the ECU (if it is an OEM ECU) from one of the clutch pedal switches that puts the ECU into idle control mode when the clutch is engaged - I believe that it blocks idle control by the ECU when the clutch is engaged. I suggest that you check operation of the clutch input to confirm that the switch has not failed and that the input is correct. Has anybody been messing around with the wiring at the clutch pedal and perhaps screwed up the connections?
Normally, I would not expect the clutch input switch failure to cause the engine to stall. I would expect that it just idles erratically or too low. However, I don't know what the supercharger does to this mix. Are you using the stock ECU with a FIC add on or do you have an aftermarket ECU? If you have an aftermarket ECU and the supercharger re-tuning effort involved changing the fuel settings then perhaps they screwed up the idle control. If you have an aftermarket ECU, perhaps there is no clutch input to check.
When you cleaned the throttle body, did you mess with the throttle position sensor at all (or did the tuning guys mess with the sensor)? The throttle position is one of the inputs to the idle control system and that might be part of the problem. However, if the engine idles fine when the car is parked and in neutral I don't think the TPS is the problem.
When shifting gears, if the engine stalls when you press the clutch pedal in and then you let the clutch pedal out after completing the shift I assume the engine restarts? If so, does the CEL light come on for a couple of seconds like it normally does at start up (if it is an OEM ECU)? If so, that indicates that the ECU lost power when you pressed the clutch pedal so you might have some kind of electrical / wiring problem to the ECU and that problem is somehow initiated by clutch operation .