Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with support Oracle for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem. Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.