Cut 3D chunk out of PCB?

Yes, it can be done. It's (often) called Z-axis milling. For my usual Chinese supplier it is a standard option and adds USD $100-$150 to a small order of 2mm thick PCBs.

You could also ask for a top layer to be routed through on a multilayer board which might give you better control over the surface finish and maybe thickness.

They use something very much like a router or end mill to cut the board outline. They can also make interior cut-outs. So the tool to do this operation is standard equipment.

I've also had controlled-depth milling done but had to be selective about the vendor to be able to get the depth tightly controlled (my project probably needed ~10 um tolerance on the depth). And it will increase cost, because they won't be able to process multiple boards in a stack; they'll have to process each panel individually.

You'll probably need to call a few vendors before you find the right one for your requirements. I also wouldn't count on getting this from a bare-bones Chinese vendor. But you'll never know until you ask.

Not a direct answer, but aimed at solving your problem: can't you stack several thin PCBs on top of each other to get the desired effect? E.g. if you want a 1.6 mm thick PCB and a 0.8 mm cut-out, you could just stack one 0.8 mm PCB with a cutout on top of a 0.8 mm PCB without one. Ordering two standard PCBs is also cheaper.