X-ray differential phase-contrast tomography is a recently-developed modality for the imaging of low-contrast biological samples. Its mathematical model is based on the first derivative of the Radon transform and the images, in practice, are reconstructed using a variant of filtered back-projection (FBP). In this paper, we develop an alternative reconstruction algorithm with the aim of reducing the number of required views, while maintaining image quality. To that end, we discretize the forward model based on polynomial B-spline functions. Then, we formulate the reconstruction as a regularized weighted-norm optimization problem with a penalty on the total variation (TV) of the solution. This leads to the derivation of a novel iterative algorithm that involves an alternation of gradient updates (FBP step) and shrinkage-thresholding (within the framework of the fast iterative shrinkage-thresholding algorithm). Experiments with real data suggest that the proposed method significantly improves upon FBP; it can handle a drastic reduction in the number of projections without noticeable degradation of the quality with respect to the standard procedure.