What is the best way to make a wall hung headboard?

Use a unique door or shutters for an easy headboard the will fasten to the wall.

Use hanging cleats.

lije this one.

You can use dimensional lumber to build a flat, stiff frame (like 1 x 2s), brace the corners on the back side. If you are going wider than a twin add a piece of framing to the center between the top and bottom and brace those connections too (divide in thirds and make 2 if you are going king). Cover the frame with something thin and light weight like underlayment. Buy foam to fit the underlayment. Buy enough polyester batting to cover the foam and fold over on the back side by 2 inches all the way around. Buy fabric big enough to cover the foam and batting and fold over by 2 inches on the back. Staple the fabric and batting to the back of the frame starting with one staple on each side, check to make sure you are not pulling too tight or hanging too loose. Continue stapling from side to side and check the front often. Fold the corners like wrapping paper on a gift and staple. Try to keep your staples within an inch of the outside edge. Trim away the excess fabric and batting. Now use two 1 x 2 boards across the back of the headboard. They should be 2 inches shorter than the width of your headboard, screw them to the frame a little down from the top and a little up from the bottom (predrill the holes so you don't split the wood). Buy a french cleat at the hardware store. Attach one side to the back of the head board on the top board and the other side to the wall. I would suggest that you add a couple of felt furniture pads to the bottom board to cushion it against the wall.

a wood door