Electrical Impedance Tomography (EIT) is a non-invasive imaging technique used to visualize the electrical properties of biological tissues by injecting boundary currents through surface electrodes and measuring the resulting voltage potentials. From a mathematical perspective, EIT is a challenging and ill-posed inverse problem. Specifically, small-scale features of the conductivity distribution may be difficult to recover from boundary measurements, and even slight perturbations in the measurements can severely impact the quality of the reconstruction. In this presentation, I will introduce a reconstruction method that integrates supervised learning with a variational regularization approach. Inspired by nonlinear compressed sensing, I will formulate a sparsity-promoting, non-smooth optimization problem, which is further enhanced by a learning-based support estimation using a suitably trained graph neural network. The key advantage of the proposed method lies in its ability to combine rigorous theoretical analysis (such as the stability of the reconstruction, and the convergence of the iterative algorithm) with compelling numerical results. Finally, I will discuss recent advances in related techniques and explore the potential for adapting the proposed approach to other nonlinear ill-posed problems.