The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. Particularly important observables are the one- and multi-point distributions of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data. This is joint work with Shirshendu Ganguly. Time permitting, we will also highlight forthcoming work with Ganguly and Lingfu Zhang developing these techniques to obtain the scaling limit of the path measure in the associated model of the continuum directed random polymer (CDRP) (as well as in the zero temperature analogue of the directed landscape), under the conditioning of large point-to-point free energy. The zero temperature case establishes a conjecture of Liu and Wang.