High-Throughput Satellite (HTS) systems with digital payload technology have been identified as a key enabler to support 5G/6G high-data connectivity with wider coverage area. The satellite community has extensively explored resource allocation methods to achieve this target. Typically, these methods do not consider the intrinsic architecture of the flexible satellite digital payload, which consists of multiple processors responsible for receiving, processing, and transmitting the signals. This paper presents a demand-aware onboard processor management scheme for broadband Non-Geostationary (NGSO) satellites. In this context, we formulate an optimization problem to minimize the number of active on-board processors while meeting the system constraints and user requirements. As the problem is non-convex, we solve it in two steps. First, we transform the problem into demand-driven bandwidth allocation while fixing the number of processors. Second, using the bandwidth allocation solution, we determine the required number of processors with two methods: 1) sequential optimization with the Branch & Bound method and 2) Bin Packing with Next Fit, First Fit, and Best Fit methods. Finally, we demonstrate the proposed methods with extensive numerical results. It is shown that the Branch & Bound, Best Fit, and First Fit methods manage the processors better than the Next Fit method. Furthermore, Branch & Bound requires fewer processors than the above methods.