Parameter estimation from noisy and one-bit quantized data has become an important topic in signal processing, as it offers low cost and low complexity in the implementation. On the other hand, Direction-of-Arrival (DoA) estimation using Sparse Linear Arrays (SLAs) has recently gained considerable interest in array processing due to their attractive capability of providing enhanced degrees of freedom. In this paper, the problem of DoA estimation from one-bit measurements received by an SLA is considered and a novel framework for solving this problem is proposed. The proposed approach first provides an estimate of the received signal covariance matrix through minimization of a constrained weighted least-squares criterion. Then, MUSIC is applied to the spatially smoothed version of the estimated covariance matrix to find the DoAs of interest. Several numerical results are provided to demonstrate the superiority of the proposed approach over its counterpart already propounded in the literature.