Finger Joints on the CNC

The simple answer is that you need to mount your boards vertically. Pulling this off requires 1) a vertical clamping fixture and 2) a CNC that allows you to mill beyond the boundary of the bed. Here are a few resources to give you the idea:

There’s much to learn about this but the results can be really beautiful.
Few software focus on this capability but JointCAM is worth checking out.
That said, the fancy stuff in the video above was all done in VCarve Pro.