This paper devises a detection architecture capable of rejecting mismatched signals embedded in Gaussian interference with unknown covariance matrix based on a sparse recovery technique. Specifically, a sparse learning method is exploited to estimate the amplitude and target angle of arrival, which are then employed to design detectors relying on the two-stage detection paradigm. Remarkably, the new decision scheme exhibits a bounded-constant false alarm rate property. The performance assessment, carried out by Monte Carlo simulations, shows that the new detectors can outperform the existing ones in terms of rejecting mismatched signals, while retaining reasonable detection performance for matched signals.