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 and singularities of the flow. By foundational work of Neves, Type I singularities are ruled out under mild assumptions, so it is important to construct examples of Type II singularities with a given blow-up model. In this talk, we describe a general method to construct examples of Lawlor neck pinching in LMCF in complex dimension at least 3. We employ a P.D.E. based approach to solve the problem, as an example of 'parabolic gluing'. The main technical tool we use is the notion of manifolds with corners and a-corners, as introduced by Joyce following earlier work of Melrose. Time permitting, we will discuss how one may construct examples of generic neck pinching.