Hey Mark! Awesome to hear that you are tackling the challenge of making your own shamisen! I am also in the process of making my own shamisen as well (not my first instrument, I have been building instruments on my own for a hobby for about 10 years now or so), so hopefully I can provide some insight to your dilemma. I think it can depend on a few different things. First off, what type of shamisen are you making? Are you making a futozao (thick neck) tsugaru shamisen, or a thinner neck one? Also, what is the current thickness of the piece of wood you cut at the deepest point?
For a tsugaru shamisen, the neck is about 1.375", or around 35mm thick, but it could be a bit thinner or thicker if you wanted. Depending on how deep your deepest point is at the cut on your neck, I think you could definitely go with the glued top like akatsuki and eclipse to fix your problem. Actually, for my own shamisen that I am working on now, this is the route I am taking, and personally, I think it is a way cooler and much better looking option than the traditional solid piece style, and makes for a more unique shamisen. If you want, you can even try and make it extend into the tenjin like eclipse/akatsuki (also what I am doing with my own), though it takes a little bit extra planning. It’s definitely not necessary at all, but is another option to consider if you go this route on your build.
I think a good balance for the topboard can be from .25" (6.35mm) to .3125" (7.94mm) thick. To give you an example from my own build, my tsugaru shamisen neck is currently 34mm thick, with the topboard wood at 8mm. You could certainly go thicker for the top wood than that, though if it is too thick it may look a bit odd proportionally, though it could be done. I think depending on how thick your current neck is now, you don’t necessarily need to start over, if you glue the extra piece on top. Also, for the topwood, you will ideally want a very hard and durable wood, such as something from the ebony or rosewood family, or other similar hardness alternative woods - there are many options available, and you can buy fretboard blanks of these materials from luthier supply shops pretty cheaply and commonly. These blanks generally run from .25" (6.35mm) thick up to .375" (9.525mm) thick. For the length of a shamisen neck, I would recommend looking at bass blanks to cover the length - usually guitar blanks aren’t long enough. However, the width will need to be reduced. Alternatively, you can also find some suppliers that sell thin stock lumber, or ask them the mill a regular board down to the thickness you want if you can’t do it yourself.
What woods are you using for your shamisen currently? You should post some pics of your work and progress! I would be more than happy to provide any additional assistance or answer whatever other questions you have for your build as well, and I am sure others here on Bachido with shamisen building experience would be glad to help out as well. Hope this helps, and looking forward to hear about the progress of your build!