In many VoIP systems, Voice Activity Detection (VAD) is often used on VoIP traffic to suppress packets of silence in order to reduce the bandwidth consumption of phone calls. Unfortunately, although VoIP traffic is fully encrypted and secured, traffic analysis of this suppression can reveal identifying information about calls made to customer service automated phone systems. Because different customer service phone systems have distinct, but fixed (pre-recorded) automated voice messages sent to customers, VAD silence suppression used in VoIP will enable an eavesdropper to profile and identify these automated voice messages. In this paper, we will use a popular enterprise VoIP system (Cisco CallManager), running the default Session Initiation Protocol (SIP) protocol, to demonstrate that an attacker can reliably use the silence suppression to profile calls to such VoIP systems. Our real-world experiments demonstrate that this side-channel profiling attack can be used to accurately identify not only what customer service phone number a customer calls, but also what following options are subsequently chosen by the caller in the phone conversation.