The large binocular telescope (LBT) can spectrally characterize faint objects from the ultraviolet (UV) to the near infrared (NIR) using two instruments, such as multiobjects double spectrograph (MODS) and LBT utility camera in the infrared (LUCI), which are pairs of imagers and spectrographs. Although LUCI can cover the NIR bands (0.9 to 2.4 μm), we currently need to use both LUCIs at the same time with existing gratings and filters. We report on the design and initial construction of a modular system called mask-oriented breadboard implementation for unscrambling spectra (MOBIUS) that enables a single LUCI to produce a full NIR spectrum (0.9 to 2.4 μm) in a single exposure. MOBIUS is a Littrow type spectrograph that is installed within the limited space of exchangeable mask frame space of LUCI. This plug-in concept requires no modification to the current instrument while dispersing the input slit perpendicular to the dispersion direction of the gratings in LUCI. With MOBIUS, we can utilize a slit length up to 2.3 arcsecond to acquire zJHK spectra without mixing orders at the LUCI image plane. In binocular observations with the LBT, a MODS spectrograph will be used with a LUCI + MOBIUS to acquire spectra across the full optical NIR wavelength range from 0.3 to 2.4 μm simultaneously. This will benefit studies of transient sources from rotating asteroids in our solar system to gamma-ray bursts, as well as anything with broad spectral features or unknown redshifts. The design process, tolerances, and initial table-top testing results to verify the operation of MOBIUS are presented in this work.