We consider slotted asynchronous channel with feedback, where the transmitter may be asleep in some slots and awake in others--the transmitter can receive perfect feedback when awake.  For the binary hypothesis testing problem where the receiver needs to decide if the transmitter is awake or not, we provide the complete characterization of the tradeoff between missed-detection and false-alarm exponents.  We then consider the problem of asynchronous data communication at rate R.  We show a simple three-phase communication strategy can achieve the best of both worlds: the maximum missed-detection exponent (while achieving vanishing false-alarm probability) and the Burnashev error exponent at data rate R for synchronous communication with feedback.