Cognitive workload affects operators' performance principally in high-risk or time-demanding situations and when multitasking is required. An online cognitive workload monitoring system can provide valuable inputs to decision-making instances, such as the operator's state of mind and resulting performance. Therefore, it can allow potential adaptive support to the operator. This work presents a new design of a wearable embedded system for online cognitive workload monitoring. This new wearable system consists of, on the hardware side, a multi-channel physiological signals acquisition (respiration cycles, heart rate, skin temperature, and pulse waveform) and a low-power processing platform. Further, on the software side, our wearable embedded system includes a novel energy-aware bio-signal processing algorithm. We also use the concept of application self-awareness to enable energy-scalable embedded machine learning algorithms and methods for online subjects' cognitive workload monitoring. Our results show that this new wearable system can continuously monitor multiple bio-signals, compute their key features, and provide reliable detection of high and low cognitive workload levels with a time resolution of 1 minute and a battery lifetime of 14.58 h in our experimental conditions. It achieves a detection accuracy of 76.6% (2.6% lower than analogous offline computer-based analysis) with a sensitivity of 77.04% and a specificity of 81.75%, on a simulated drone rescue mission task. Moreover, by applying our self-aware monitoring to exploit different energy-scalable modes, we can increase battery lifetime by 51.6% (up to 22.11 hours) while incurring an insignificant accuracy loss of 1.07%.