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This paper proposes an evolutionary transfer learning approach (Evol-TL) for scalable quality-of-transmission (QoT) estimation in multi-domain elastic optical networks (MD-EONs). Evol-TL exploits a broker-based MD-EON architecture that enables cooperative learning between the broker plane (end-to-end) and domain-level (local) machine learning functions while securing the autonomy of each domain. We designed a genetic algorithm to optimize the neural network architectures and the sets of weights to be transferred between the source and destination tasks. We evaluated the performance of Evol-TL with three case studies considering the QoT estimation task for lightpaths with (i) different path lengths (in terms of the numbers of fiber links traversed), (ii) different modulation formats, and (iii) different device conditions (emulated by introducing different levels of wavelength-specific attenuation to the amplifiers). The results show that the proposed approach can reduce the average amount of required training data by up to $13\times <\#comment/>$while achieving an estimation accuracy above 95%.