The GENERIC framework provides a robust structure for nonequilibrium dynamics but lacks a principled method to select reversible (L) and irreversible (M) brackets. Similarly, f inite-time optimizations minimizing path-averaged reciprocal temperature exist but remain isolated. Here, we introduce the Entropy Variational Law (EVL)—a principle elevating minimization of (dS/dE)dE to a guiding rule for nonequilibrium evolution. Embedded in GENERIC, EVL yields gGENERIC, prescribing dynamics via a concrete Poisson–EVL splitting scheme in which the irreversible step is obtained from a local entropy-variational quadratic program. At the continuum level, gGENERIC recovers the standard structure of irreversible thermodynamics and provides a constructive route to bracket selection. Model problems illustrate how gGENERIC reproduces known optimal heating protocols, standard thermodynamic laws, and black-hole-inspired entropy balances, suggesting its usefulness as a flexible building block for entropy-based modelling of nonequilibrium processes in physics and chemistry.