Hole partially through PCB
This is something which is usually not covered by the functionality of common EDA-Tools like EAGLE. 

Here's why:
This is done by milling after processing all other production steps of the PCB. It can't be done by just omitting something in the prepregs used, as they can contain only drills for vias. And the latter is only done when buried vias are needed.

Milling through some of the layers is rarely needed, so there is no standard in handling those special requests. It can't be defined in gerber or drill files. There is no need to describe it in an EDA tool, as this doesn't make any sense, when no CAM file can't be generated to describe the necessary milling actions.

There is a milling layer in EAGLE, but it it's only purpose are millings through all layers of the PCB.

You might need some luck to find a PCB manufacturer who is even able to do such millings. 

I recommend to add a new layer for each milling depth you want to have on your PCB and export it as DXF along with the alignment marks. Then explain your demands to your favourite PCB manufacturer and see what happens.

Edit: There are further arguments, why EAGLE doesn't provide mechanism to describe intricate millings. Most layers describe electrical features and the copper layers can hold information only on the distinction where to remove copper and where not. There are no layers describing the prepregs and where they should have gaps or not.