Both sides don't have to be the same thickness.

Do the back side first. If there are any errors, glitches, problems, etc., you will identify and correct them before working on the face side.

Do a test run with a small piece of your walnut on some scrap. Check to make sure the glue doesn't penetrate through the veneer, to the surface. Reduce the amount of glue you apply, if it does penetrate to the surface. Since you've done this before, I would think you won't have a penetration issue, but it's always good to do a test run.

*If all goes bad, you can always upholster the headboard. LOL.

Sonny