Reconstructing an avatar from a portrait image has many applications in multimedia, but remains a challenging research problem. Extracting reflectance maps and geometry from one image is ill-posed: recovering geometry is a one-to-many mapping problem, and reflectance and light are difficult to disentangle. Accurate geometry and reflectance can be captured under the controlled conditions of a light stage, but it is costly to acquire large datasets in this fashion. Moreover, training solely with this type of data leads to poor generalization with in-the-wild images. This motivates the introduction of MoSAR, a method for 3D avatar generation from monocular images. We propose a semi-supervised training scheme that improves generalization by learning from both light stage and in-the-wild datasets. This is achieved using a novel differentiable shading formulation. We show that our approach effectively disentangles the intrinsic face parameters, producing relightable avatars. As a result, MoSAR estimates a richer set of skin reflectance maps and generates more realistic avatars than existing state-of-the-art methods. We also release a new dataset, that provides intrinsic face attributes (diffuse, specular, ambient occlusion and translucency maps) for 10k subjects. Project, code and dataset: https://ubisoft-laforge.github.io/character/mosar