“Does the air traffic control radar have the capability to determine the debris is moving at supersonic speed?”
Yes, assuming the debris is within the range and view of the radar.
I would think the only way ATC radar could determine the speed of debris is if the debris is moving directly towards or away from the radar, and it monitored the Doppler shift of the radar return, and even then only if the explosion occurred immediately before the radar painted the target.
I wouldn’t think ATC radars would do speed determination via doppler shift; that would only show speed relative to the radar station. Speed would be actually determined by a dx/dt calculation.