We report on the internal configurations of droplets of nematic liquid crystals (LCs; 10-50 mu m-in-diameter; comprised of 4-cyano-4'-pentylbiphenyl and 4-(3-acryloyloxypropyloxy) benzoic acid 2-methyl-1,4-phenylene ester) sedimented from aqueous solutions of sodium dodecyl sulfate (SDS) onto interfaces formed with pure glycerol. We observed a family of internal LC droplet configurations and topological defects consistent with a remarkably abrupt transition from homeotropic (perpendicular) to tangential anchoring on the surface of the LC droplets in the interfacial environment. Calculations of the interdiffusion of water and glycerol at the aqueous-glycerol interface revealed the thickness of the diffuse interfacial region of the two miscible liquids to be small (0.2-0.5 mu m) compared to the diameters of the LC droplets on the experimental time-scale (15-120 minutes), leading us to hypothesize that the patterned surface anchoring was induced by gradients in concentration of SDS and glycerol across the diameter of the LC droplets in the interfacial region. This hypothesis received additional support from experiments in which the time of sedimentation of the LC droplets onto the interface was systematically increased and the droplets were photo-polymerized to preserve their configurations: the configurations of the LC droplets were consistent with a time-dependent decrease in the fraction of the surface area of each droplet exhibiting homeotropic anchoring. Specifically, LC droplets with <10% surface area with tangential anchoring exhibited a bulk point defect within the LC droplet, whereas droplets with <10% surface area with tangential anchoring exhibited a boojum defect within the tangential region and a disclination loop separated the regions with tangential and homeotropic anchoring. The topological charge of these LC droplet configurations was found to be consistent with the geometrical theorems of Poincare and Gauss and also well-described by computer simulations performed by minimization of a Landau-de Gennes free energy. Additional experimental observations (e.g., formation of "Janus-like'' particles with one hemisphere exhibiting tangential anchoring and the other perpendicular anchoring) and simulations (e.g., a size-dependent set of LC droplet configurations with <10% surface area exhibiting tangential anchoring) support our general conclusion that placement of LC droplets into miscible liquid-liquid interfacial environments with compositional gradients can lead to a rich set of LC droplet configurations with symmetries and optical characteristics that are not encountered in LC droplet systems in homogeneous, bulk environments. Our results also reveal that translocation of LC droplets across liquid-liquid interfaces can define new transition pathways that connect distinct configurations of LC droplets.