Machine Unlearning has recently been emerging as a paradigm for selectively removing the impact of training datapoints from a network. While existing approaches have focused on unlearning either a small subset of the training data or a single class, in this paper we take a different path and devise a framework that can unlearn all classes of an image classification network in a single untraining round. Our proposed technique learns to modulate the inner components of an image classification network through memory matrices so that, after training, the same network can selectively exhibit an unlearning behavior over any of the classes. By discovering weights which are specific to each of the classes, our approach also recovers a representation of the classes which is explainable by-design. We test the proposed framework, which we name Weight Filtering network (WF-Net), on small-scale and medium-scale image classification datasets, with both CNN and Transformer-based backbones. Our work provides interesting insights in the development of explainable solutions for unlearning and could be easily extended to other vision tasks.