In Extreme Multi Label Completion (XMLCo), the objective is to predict the missing labels of a collection of documents. Together with XML Classification, XMLCo is arguably one of the most challenging document classification tasks, as the very high number of labels (at least ten of thousands) is generally very large compared to the number of available labelled documents in the training dataset. Such a task is often accompanied by a taxonomy that encodes the labels organic relationships, and many methods have been proposed to leverage this hierarchy to improve the results of XMLCo algorithms. In this paper, we propose a new approach to this problem, TAMLEC (Taxonomy-Aware Multi-task Learning for Extreme multi-label Completion). TAMLEC divides the problem into several Taxonomy-Aware Tasks, i.e. subsets of labels adapted to the hierarchical paths of the taxonomy, and trains on these tasks using a dynamic Parallel Feature sharing approach, where some parts of the model are shared between tasks while others are task-specific. Then, at inference time, TAMLEC uses the labels available in a document to infer the appropriate tasks and to predict missing labels. To achieve this result, TAMLEC uses a modified transformer architecture that predicts ordered sequences of labels on a Weak-Semilattice structure that is naturally induced by the tasks. This approach yields multiple advantages. First, our experiments on real-world datasets show that TAMLEC outperforms state-of-the-art methods for various XMLCo problems. Second, TAMLEC is by construction particularly suited for few-shots XML tasks, where new tasks or labels are introduced with only few examples, and extensive evaluations highlight its strong performance compared to existing methods.