Knowledge of materials' thermal-transport properties, conductivity and diffusivity, is crucial for several applications within areas of biology, material science and engineering. Specifically, a microsized, flexible, biologically integrated thermal transport sensor is beneficial to a plethora of applications, ranging across plants physiological ecology and thermal imaging and treatment of cancerous cells, to thermal dissipation in flexible semiconductors and thermoelectrics. Living cells pose extra challenges, due to their small volumes and irregular curvilinear shapes. Here a novel approach of simultaneously measuring thermal conductivity and diffusivity of different materials and its applicability to single cells is demonstrated. This technique is based on increasing phonon-boundary-scattering rate in nanomembranes, having extremely low flexural rigidities, to induce a considerable spectral dependence of the bandgap-emission over excitation-laser intensity. It is demonstrated that once in contact with organic or inorganic materials, the nanomembranes' emission spectrally shift based on the material's thermal diffusivity and conductivity. This NM-based technique is further applied to differentiate between different types and subtypes of cancer cells, based on their thermal-transport properties. It is anticipated that this novel technique to enable an efficient single-cell thermal targeting, allow better modeling of cellular thermal distribution and enable novel diagnostic techniques based on variations of single-cell thermal-transport properties.