An assembly is provided for a turbine engine. This turbine engine assembly includes a combustor wall including a first layer vertically connected with a second layer. A first portion of the first layer overlaps and is vertically spaced from the second layer by a cavity. A second portion of the first layer is substantially vertically inline with an adjacent portion of the second layer. The second portion of the first layer at least partially forms a quench aperture vertically through the combustor wall.