This paper introduces a novel optimization approach for stain separation in digital histopathological images. Our stain separation cost function incorporates a smooth total variation regularization and is minimized by using a projected gradient algorithm. To enhance computational efficiency and enable supervised learning of the hyperparameters, we further unroll our algorithm into a neural network. The unrolled architecture is not only more efficient for solving the stain separation problem, but also allows to design a highly interpretable and flexible method. Experimental results demonstrate the effectiveness of the proposed unrolled projected gradient algorithm in achieving accurate and visually consistent stain separation.