The Unmanned Aerial Vehicle (UAV) has recently become a cost-effective and flexible data collection tool for Wireless Sensor Networks (WSNs). However, UAVs are typically energy-constrained with limited battery capacity, so their hover positions and flight trajectory during the data collection journey require careful planning to be efficient to have longer serving time and more coverage. In addition, reconfigurable antennas can provide adjustable coverage area for UAVs, which can further improve the efficiency of data collection. In order to address the efficient WSN data collection problem using a UAV equipped with reconfigurable antenna, in this paper, we propose BROAD, an iterative algorithm that can complete all data collection under sensor energy constraints while minimizing the total UAV energy consumption. Due to the non-convexity of the data collection problem, to determine the UAV hovering position, antenna beamwidth, and UAV travel path, we decompose the original problem into three sub-problems. We first use the linearized objective function to jointly optimize hover positions and beamwidth, then determine the covered sensor allocation. Finally, we use the travel salesman model to determine the UAV visit order. Through extensive simulations comparing with two benchmark algorithms under various parameters, BROAD manages to provide 27.5% more energy savings, which is very close to the theoretical lower bound.