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We present the Migdal In Galactic Dark mAtter expLoration (MIGDAL) experiment aiming at the unambiguous observation and study of the so-called Migdal effect induced by fast-neutron scattering. It is hoped that this elusive atomic process can be exploited to enhance the reach of direct dark matter search experiments to lower masses, but it is still lacking experimental confirmation. Our goal is to detect the predicted atomic electron emission which is thought to accompany nuclear scattering with low, but calculable, probability, by deploying an Optical Time Projection Chamber filled with a low-pressure gas based on CF$_4$. Initially, pure CF$_4$ will be used, and then in mixtures containing other elements employed by leading dark matter search technologies -- including noble species, plus Si and Ge. High resolution track images generated by a Gas Electron Multiplier stack, together with timing information from scintillation and ionisation readout, will be used for 3D reconstruction of the characteristic event topology expected for this process -- an arrangement of two tracks sharing a common vertex, with one belonging to a Migdal electron and the other to a nuclear recoil. Different energy-loss rate distributions along both tracks will be used as a powerful discrimination tool against background events. In this article we present the design of the experiment, informed by extensive particle and track simulations and detailed estimations of signal and background rates. In pure CF$_4$ we expect to observe 8.9 (29.3) Migdal events per calendar day of exposure to an intense D-D (D-T) neutron generator beam at the NILE facility located at the Rutherford Appleton Laboratory (UK). With our nominal assumptions, 5$σ$ median discovery significance can be achieved in under one day with either generator.