This paper presents a Large Eddy Simulation (LES) methodology coupled with the Doubly Conditional Moment Closure (DCMC) model and detailed chemistry for investigating dual-fuel combustion problems. The approach uses both mixture fraction and reaction progress variable as conditioning variables to characterize the mixing and thermochemical history of different equivalence ratios. The approach is then tested against a canonical turbulent jet ignition problem, achieved via a newly developed DCMC solver embedded within the CONVERGE CFD software. Examination of the solutions provide insight into the physical and conserved scalar space for this case. The approach is shown to capture well all relevant combustion modes, i.e. premixed flame propagation, non-premixed ignition, and partially premixed propagation. The results demonstrate a comprehensive framework in characterizing dual-fuel combustion and can help with the design of future combustors and engines relying on dual-fuel and multi-mode combustion.