An assembly for a turbine engine includes a combustor wall. The combustor wall includes a shell, a heat shield and an annular land. The heat shield is attached to the shell. The land extends vertically between the shell and the heat shield. The land extends laterally between a land outer surface and an inner surface, which at least partially defines a quench aperture in the combustor wall. A lateral distance between the land outer surface and the inner surface varies around the quench aperture.