The widespread adoption of residential electric vehicles (EVs) presents an untapped opportunity to enhance the resilience of distribution networks during emergencies. Private EV chargers have the potential to support critical loads during network emergencies by leveraging advanced communication infrastructure in smart grids. This paper proposes a novel optimization framework to harness this potential, addressing the complexities of network restoration and dimensionality. The developed multi-stage, multi-objective, and rolling horizon framework integrates the equivalent storage capacity of residential EVs into the distribution network restoration process while considering uncertainties and detailed modelling aspects. To do this, first, the aggregators calculate the equivalent storage capacity of all participating EVs and submit the aggregate proposal to the network operator in a rolling horizon manner to account for unforeseen changes and forecasting deviations. Then, the network operator schedules the received equivalent storage capacities in lexicographically ordered objectives to maximize recovery rates of critical loads while minimizing associated costs. The charging and discharging commitments are then returned to the aggregators to be distributed among the participating EVs via predefined rules. The test case results and the model's linear nature ensure its practical applicability for large-scale and real-life distribution networks.