We construct a finite-time blow-up solution to the mass-critical focusing nonlinear Schrödinger equation on a metric star graph with an arbitrary number of edges. We show that all solutions are global if their mass is smaller than an explicit constant, called ``minimal mass''. We then construct a solution with minimal mass and arbitrary energy, which blows up in finite time at the vertex of the star graph. The blow-up profile and blow-up speed are explicitly characterized. The main novelty of the paper is the construction of the blow-up profile in time-dependent domains of singularly perturbed Laplacians.