Infrared devices are increasingly used in industrial, medical, and environmental monitoring applications. Cost-effectiveness, robustness, and portability are characteristics that are highly sought after and they can be enabled by a dispersive spectrometer carrying a single-pixel detector. In this paper, we demonstrate a novel, high-throughput dispersive spectrometer that has its spectral resolution decoupled from its throughput. The proposed spectrometer implements a two-stage Hadamard transform encoding process that allows significantly more light into the system to enhance its signal-to-noise ratio. As a single-pixel detector is used to collect the spectral information, the proposed system can be easily implemented in other desired wavelengths. Furthermore, we develop a method to remove the need for uniform illumination at the entrance aperture by taking into consideration its spatial information during the reconstruction process, thereby increasing the ease of the design of devices required for in situ measurement.