The combination of network softwarization with network slicing enables the provisioning of very diverse services over the same network infrastructure. However, it also creates a complex environment where the orchestration of network resources cannot be guided by traditional, human-in-the-loop network management approaches. New solutions that perform these tasks automatically and in advance are needed, paving the way to zero-touch network slicing. In this paper, we propose AZTEC, a data-driven framework that effectively allocates capacity to individual slices by adopting an original multi-timescale forecasting model. Hinging on a combination of Deep Learning architectures and a traditional optimization algorithm, AZTEC anticipates resource assignments that minimize the comprehensive management costs induced by resource overprovisioning, instantiation and recon-figuration, as well as by denied traffic demands. Experiments with real-world mobile data traffic show that AZTEC dynamically adapts to traffic fluctuations, and largely outperforms state-of-the-art solutions for network resource orchestration.