Lagrangian mean curvature flow (LMCF) is a way to deform a Lagrangian submanifold inside a Calabi–Yau manifold according to the negative gradient of the area functional. There are influential conjectures about LMCF due to Thomas–Yau and Joyce, describing the long-time behaviour of the flow, singularity formation, and how one may flow past singularities. In this talk, we will show how one may flow out of a conically singular Lagrangian by gluing in expanders asymptotic to the cone, generalizing an earlier result by Begley–Moore. We solve the problem by a direct P.D.E.-based approach, along the lines of recent work by Lira–Mazzeo–Pluda–Saez on the network flow. The main technical ingredient we use is the notion of manifolds with corners and a-corners as introduced by Joyce following earlier work of Melrose.