The Smart Grid is expected to utilize a wireless infrastructure for power data collection in its Advanced Metering Infrastructure (AMI) applications. One of the options to implement such a network infrastructure is to use a wireless mesh network based on IEEE 802.11s mesh standard. However, IEEE 802.11s standard relies on MAC-based routing and thus requires the availability of MAC addresses of destinations. Due to large size of AMI networks, this creates a broadcast storm problem when such information is to be obtained via Address Resolution Protocol (ARP) broadcast packets. In this paper, we propose a mechanism to significantly alleviate such broadcast storm problem in order to improve the scalability of 802.11s and thus make it better suited for Smart Grid AMI applications. Our contribution is adapting 802.11s standard for addressing ARP broadcast storm problem in a secure and efficient manner. Specifically, we utilize the proactive Path Request (PREQ) packet and Path Reply (PREP) of layer-2 path discovery protocol of 802.11s, namely HWMP, for piggybacking ARP information. In this way, the MAC address resolution is handled during routing tree creation/maintenance and hence the broadcasting of ARP requests by the smart meters (SMs) to learn the MAC address of the data collector (i.e., the gateway/root node) is completely eliminated. Furthermore, since piggybacking the ARP via PREQ may pose vulnerabilities for possible ARP cache poisoning attacks, the data collector also authenticates the messages it sends to SMs by using Elliptic Curve Digital Signature Algorithm (ECDSA). We have extensively analyzed the behavior and overhead of the proposed mechanism using implementation of IEEE 802.11s in ns-3 simulator. The evaluations for both UDP and TCP show that compared to the original ARP broadcast operations, our approach reduces the end-to-end delay significantly without negatively impacting the packet delivery ratio and throughput.