Oscillatory rheometric techniques are widely used for rheological characterization of the viscoelastic properties of complex fluids. However, in a mutating material the evolution of microstructure is commonly both time– and shear-rate–dependent, and thixotropic phenomena are observed in many complex fluids. Application of the Fourier transform for analyzing oscillatory data implicitly assumes the signals are time-translation invariant which constrains the mutation number of the sample to be extremely small. This constraint makes it difficult to accurately study shear-induced microstructural changes in thixotropic and gelling materials.  In this work, we explore applications of the Gabor transform (a Short Time Fourier Transform (STFT) combined with a Gaussian window) for providing optimal time-frequency resolution of the viscoelastic properties of a mutating material. First, we show using simple models that application of the STFT enables extraction of useful data from the initial transient response following the inception of oscillatory flow.  Secondly, using measurements on a Bentonite clay, we show that a Gabor transform enables us to more accurately measure rapid changes in the complex modulus and also extract a characteristic thixotropic/aging time scale for the material.  Finally, we consider extension of the Gabor transform to non-linear oscillatory deformations using an amplitude-modulated input signal, in order to track the evolution of the Fourier-Tschebyshev coefficients characterizing a thixotropic fluid. We refer to the resulting test protocol as Gaborheometry and construct an operability diagram in terms of the imposed ramp rate and the mutation time of the material. By considering the shapes of Gabor spectrograms we show that there is a trade-off between frequency and time resolution (effectively a rheological uncertainty principle!). This unconventional, but easily implemented, rheometric approach facilitates both SAOS and LAOS studies of time-evolving materials, reducing the number of experiments and the data post-processing time significantly.