Either a router or a Forstner (or similar) drill bit are both good ways to make the recesses you want, but in a small piece of wood I think a drill bit is clearly the better option. If you can find a suitable flat bit/spade bit cheaply you might also like to try this proposed modification from a previous Answer.
Regardless of the method used to mill the recess your main difficulty is in holding such small pieces so that the wood is secure and the operation is safe, so that's the thing to concentrate on.
With a router you'd need to build a custom clamping rig with jaws that were the same thickness as your slices or slightly thinner, and additionally the wood needs to be held tighter because of the very high speeds that routers run at.
It's much simpler with a drill bit since the clamping device or aid can be thicker than the workpiece (much thicker in fact because of the long shaft on the bit) and it won't get in the way.
One of the standard ways woodworkers hold small parts is with a handscrew:
These both show routing operations but you can see how you could just as easily do the same for drilling.
You probably don't own a handscrew and although a simple version is easy enough to make at home a number of other clamp types are capable of holding the work in a similar way. F-clamps, quick clamps and even a shorter sash clamp could all be used for this without any problem.
Personally I think you could safely hold the workpiece by hand if you have a drill stand or pillar drill, but if you want to be more careful than that another trick would be to attach the slices to a long piece of wood that you can clamp firmly off to one side or hold by hand with your fingers safely away from the spinning bit.
There are various ways to temporarily bond one piece of wood to another, and here double-sided tape may work perfectly well. But if you find it's not secure enough I suggest you try using superglue between two strips of tape as I describe in How do I temporarily attach two pieces of wood together for machining?