We propose min Generalized Sliced Gromov--Wasserstein (min-GSGW), a sliced formulation for the Gromov--Wasserstein (GW) problem using expressive generalized slicers. The key idea is to learn coupled nonlinear slicers that assign compatible push-forward values to both input measures, so that monotone coupling in the projected domain lifts to a transport plan evaluated against the GW objective in the original spaces. The resulting plan induces a GW objective value, and min-GSGW minimizes this cost directly in the original spaces. We further show that min-GSGW is rigid-motion invariant, a crucial property for geometric matching and shape analysis tasks. Our contributions are threefold: 1) we introduce generalized slicers into the sliced GW framework, 2) we construct a slicing-based efficient GW transport plan; and 3) we develop an amortized variant that replaces per-instance optimization with a learned slicer for unseen input pairs. We perform experiments on animal mesh matching, horse mesh interpolation, and ShapeNet part transfer. Results show that min-GSGW produces meaningful geometric correspondences and GW objective values at substantially lower computational cost than existing GW solvers.