I build mountain dulcimers & I cut a silhouette through the scrollhead, which is about 2.5"-3" thick. I currently use a scrollsaw for that, which is a total nightmare! I get burned wood, broken blades, injuries, all that fun stuff. I'm looking to see if there are better alternatives.

What I know about CNC machines could fit on a postage stamp, so I'm asking the community:
Can a CNC cut through 3" of hard wood?
Can a CNC cut triangular points, such as stars have? (I've only seen round shapes done so far.)
What type (if any) machine would do that job?

Any input, answers, or direction would be greatly appreciated!