Estimating a high-dimensional sparse covariance matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, we consider estimating a sparse shape matrix from n samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We prove that under suitable conditions the regularized variant can be computed efficiently in practical polynomial time. Furthermore, we prove that in the joint limit as dimension p and sample size n tend to infinity with p/n tending to a constant, our proposed estimators are minimax rate optimal. Results on simulated data support our theoretical analysis.