In this talk, I will present the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. A key focus of the talk will be the surgery method we developed to handle these singularities and establish global asymptotics. I will also discuss applications of this result, including the convergence of the characteristic polynomial of random normal matrices to Gaussian Multiplicative Chaos measure. Based on joint work with Paul Bourgade, Guillaume Dubach, and Lisa Hartung.