A hallmark of far-from-equilibrium physics is the emergence of a spectral cascade, where energy is transferred across length-scales following a simple power law. Scaling laws of steady states have been successfully predicted in fluid dynamics by the statistical theory of weak wave turbulence. However, the predictive power of this theory breaks down in presence of large amplitudes, high dissipation, and finite size effects. We offer experimental insight into these regimes by resolving the dynamics of individual wave modes in an externally driven fluid-fluid interface. We observe the time evolution of interface excitations from one to few to many, a process culminating in a direct energy cascade. Our findings confirm that interfacial dynamics can be effectively modelled by a weakly nonlinear Lagrangian theory, revealing a hierarchy in their order that confirms a key assumption of weak wave turbulence. Specific interactions are tracked through time, and we predict the timescale until a cascade emerges. Furthermore, we highlight how our experiment may inform us about other far-from-equilibrium systems by mapping our Lagrangian theory to a model of cosmological preheating in the early universe.