Sparse continuous policies are distributions that can choose some actions at random yet keep strictly zero probability for the other actions, which are radically different from the Gaussian. They have important real-world implications, e.g. in modeling safety-critical tasks like medicine. The combination of offline reinforcement learning and sparse policies provides a novel paradigm that enables learning completely from logged datasets a safety-aware sparse policy. However, sparse policies can cause difficulty with the existing offline algorithms which require evaluating actions that fall outside of the current support. In this paper, we propose the first offline policy optimization algorithm that tackles this challenge: Fat-to-Thin Policy Optimization (FtTPO). Specifically, we maintain a fat (heavy-tailed) proposal policy that effectively learns from the dataset and injects knowledge to a thin (sparse) policy, which is responsible for interacting with the environment. We instantiate FtTPO with the general $q$-Gaussian family that encompasses both heavy-tailed and sparse policies and verify that it performs favorably in a safety-critical treatment simulation and the standard MuJoCo suite. Our code is available at \url{https://github.com/lingweizhu/fat2thin}.