Minimal Lagrangian tori in CP^{2} are the expected local model for particular point singularities of Calabi-Yau 3-folds and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods we construct new examples of minimal Lagrangian surfaces of genus 1/2(k-1)(k-2) for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit.