This paper proposes an optimization framework for power and time resource allocation during time sharing non-orthogonal multiple access (TS-NOMA) transmissions performed by an unmanned aerial vehicle (UAV) in the context of a large-scale scenario. The objective of the proposed UAV-TS-NOMA system and optimization framework is to jointly maximize the energy efficiency (EE) and the downlink throughput fairness among users within the UAV communication range. The idea behind is to propose a communication system that: i) merges the advantages of UAV communications with the ones offered by the TS-NOMA paradigm and ii) maximizes the EE and the downlink fairness among users. The resulting model finds applicability in performing energy efficient and throughput fair transmissions into power-constrained communication scenarios. Performance investigations regarding the proposed framework in finding the optimal set of resources which maximizes jointly the above mentioned network metrics, have shown the advantage of the proposed two-step optimization framework in finding the optimal configuration of both power and time resources, respecting both the power constraints at the transmitter and the quality-of-service requirement of the users. In addition, it is shown how under particular conditions the proposed framework jointly optimizes the aforementioned network metrics in only one step.