I doubt a basic could, but I don’t know. With repeated passes and changing the focus perhaps. You could stack thinner sections to achieve greater depth. An Acrylic solvent fuses the stuff very well.
Careful with dumping a lot of heat into acrylic (going slow with high power), we have seen at least two machines burned down because of it and the operator wandering off. Nothing to be paranoid about, just stand by with a damp rag you can drop on a flair-up.
I’ve thought about using the solvent to bond two layers, but with the thinness of the walls of the cookie cutter I wasn’t sure if I would be able to line them up well enough to use the solvent. I guess I’ll need to experiment a bit more to test it out… maybe make a jig out of wood that can hold the shape between the two layers while I use the solvent to fuse them…
I cut 1/2 inch acrylic. It took four passes and I had to set up a custom support for it. What I found is that the acrylic holds so much heat inside that it will warp and start to smolder. If you want to cut it, plan on multiple passes, each with a wait time for the material to stabilize. You may still get movement. The thinner the area you want to retain the more it will move. On the part I did, anything thinner than 1/4 inch was moving all over the place to the point where sometimes the second pass or third pass was hitting material that had moved into the kerf.
So yes, you can do it, but don’t expect it to turn out perfectly the first time and be VERY careful.
One thing I am wondering if if instead of making a cutter to make something you can compress into a pocket rather than cutting. Alternatively if you used 3/8 to 1/2" maple you could achieve the same goal that would be stronger and just as able to be made food safe with no chance of melting.