A new double time-of-flight (dTOF) neutron spectroscopy technique has been developed for pulsed broad spectrum sources with a duty cycle that results in frame overlap, where fast neutrons from a given pulse overtake slower neutrons from previous pulses. Using a tunable beam at the 88-Inch Cyclotron at Lawrence Berkeley National Laboratory, neutrons were produced via thick-target breakup of 16 MeV deuterons on a beryllium target in the cyclotron vault. The breakup spectral shape was deduced from a dTOF measurement using an array of EJ-309 organic liquid scintillators. Simulation of the neutron detection efficiency of the scintillator array was performed using both GEANT4 and MCNP6. The efficiency-corrected spectral shape was normalized using a foil activation technique to obtain the energy-dependent flux of the neutron beam at zero degrees with respect to the incoming deuteron beam. The dTOF neutron spectrum was compared to spectra obtained using HEPROW and GRAVEL pulse height spectrum unfolding techniques. While the unfolding and dTOF results exhibit some discrepancies in shape, the integrated flux values agree within two standard deviations. This method obviates neutron time-of-flight spectroscopy challenges posed by pulsed beams with frame overlap and opens new opportunities for pulsed white neutron source facilities.