Condensed matter in the low temperature limit reveals much exotic physics associated with unusual orders and excitations, examples ranging from helium superfluidity to magnetic monopoles in spin ice. The far-from-equilibrium physics of such low temperature states may be even more exotic, yet to access it in the laboratory remains a challenge. Here we demonstrate a simple and robust technique, the 'magnetothermal avalanche quench', and its use in the controlled creation of nonequilibrium populations of magnetic monopoles in spin ice at millikelvin temperatures. These populations are found to exhibit spontaneous dynamical effects that typify far-from-equilibrium systems, yet are captured by simple models. Our method thus opens the door to the study of far-from-equilibrium states in spin ice and other exotic magnets.