My name is Bert Rabbe and I am the Special Interest Group leader for the Multicam CNC Router. Our CNC Router is a 3 axis machine (X,Y,Z) and to make the examples in your photo, you would have to be sure all the paths can be reached with a bit that can only move in a 90 degree angle from the table. As an example, it the inner edges of the mustache can be carved with a (user supplied) small ball tipped tappered spiral bit of this style:
the you should be able to carve them. Building the models is a bit of another issue. You would either have to use one of the DMS supplied modeling programs like Fusion 360 or other options if you do not have your own modeling capability. Processing the model to a machine path should be quite doable.
Hope this helps. Look me up when you join.