You will likely need to do three runs.  One to do the cut outs of the birds, the other engraved and the last the frame for the box.  Process the three canvases.

Place glue at the joints for the frame.  Once the frame is glued, add a thin layer of glue around the outside of the engraved piece.  Place the frame on engraved piece.  Place thin glue on the bottom of the cutout piece and place on top of the frame.  Remove excess glue and let dry.

These pieces can stand on their own if you want them on a counter.