ABSTRACT We present a novel, purely geometric probe of cosmology based on measurements of differential time delays between images of strongly lensed quasars due to finite source effects. Our approach is solely dependent on cosmology via a ratio of angular diameter distances, the image separation, and the source size. It thereby entirely avoids the challenges of lens modelling that conventionally limit time delay cosmography, and instead entails the lensed reverberation mapping of the quasar broad-line region. We demonstrate that differential time delays are measurable with short-cadence spectroscopic monitoring of lensed quasars, through the timing of kinematically identified features within the broad emission lines. This provides a geometric determination of an angular diameter distance ratio complementary to standard probes, and as a result is a potentially powerful new method of constraining cosmology.