We present a fully three-dimensional kinetic framework for modeling intense short pulse lasers interacting with dielectric materials. Our work modifies the open-source Particle-In-Cell (PIC) code EPOCH to include new models for photoionization and dielectric optical response. We use this framework to model the laser-induced damage of dielectric materials by few-cycle laser pulses. The framework is benchmarked against experimental results for bulk silica targets and then applied to model multi-layer dielectric mirrors with a sequence of simulations with varying laser fluence. This allows us to better understand the laser damage process by providing new insight into energy absorption, excited particle dynamics, and nonthermal excited particle distributions. We compare common damage threshold metrics based on the energy density and excited electron density.