We demonstrate that nonlinear Raman spectroscopy coupled with aerosol optical tweezers can be used to probe the evolving phase partitioning in mixed organichnorganic/aqueous aerosol droplets that adopt a core-shell structure in which the aqueous phase is coated in an organic layer. Specifically, we demonstrate that the characteristic fingerprint of wavelengths at which stimulated Raman scattering is observed can be used to assess the phase behavior of multiphase decane/aqueous sodium chloride droplets. Decane is observed to form a layer on the surface of the core aqueous droplet, and from the spectroscopic signature the aqueous core size can be determined with nanometer accuracy and the thickness of the decane layer with an accuracy of +/- 8 nm. Further, the presence of the organic layer is observed to reduce the rate at which water evaporates from the core of the droplet with an increasing rate of evaporation observed with diminishing layer thickness.