Well you could sew it using a commercial machine ,I have a friend that makes early lincoln interiors that does do that by sewing the backside through the luan and rolling it over to glue to the back side of the board. But for a home user I'd just cut a piece of material as long as needed to go around where you want to cover and about 2 inches wide then place it on the board/carpet backside facing out and stapling it to the luan, after completely stapling it to the board you would then Roll the material over the edge and to the back side giving you a finished look. the either staple or glue it to the back side of the board. (heck it's harder to explain it than just do it imagine it looking like this
View attachment 1714854574
the closer you do the hidden staples on the front side the smoother the roll over looks, don't use really long staples for either side or it will go through the material.