Free-electron radiation, such as Cherenkov radiation and transition radiation, can generate light at arbitrary frequencies and is fundamental to diverse applications, ranging from electron microscopy, spectroscopy, lasers, to particle detectors. Generally, the features of free-electron radiation are stochastic when electrons interact with random media. Counterintuitively, here, we reveal a type of free-electron radiation that has both its intensity and directionality invariant to specific sorts of long-range structural randomness. Essentially, this invariance is enabled by the Brewster effect and the judiciously engineered phase coherence condition of emitted light, namely that the light induced by electron's penetration through a layered aperiodic nanostructure is engineered to interfere constructively at the Brewster angle. As such, when each constituent layer with a random thickness fulfills this phase coherence condition, there is always the emergence of free-electron resonance transition radiation at the Brewster angle. At this resonant Brewster angle, we further find that the radiation intensity and directionality could be enhanced by orders of magnitude by readily increasing the interface number. The revealed resonance transition radiation via long-range Brewster randomness may offer a feasible route to explore more enticing photonic applications driven by free electrons, such as light sources at previously unreachable spectral regimes, optical frequency combs, particle detectors, and random lasers.