I'm in need of making about a 5/16 counter bore into hard wood only about
1/16th deep. Exact measurements are not needed but it should be fairly flat
with no pilot hole(or something that is no more than 1/32nd deep).

I can use a press and again, position and hole size are not that important.
Basically I am trying to create space so that some woot can fit flush with,
say, a screw head, but the wood is only about 1/8 thick(so any pilot holes
may go completely through).

Actually I would like for the holes to be square or rectangle(as I have some
other objects I need to cut "slots" in the board that do not go all the way
through.

Accuracy is not important except for depth since I cannot go through the
board and I would like it to be relatively nice looking cuts. Circular bores
have to be oversized because the things I am using are actually square or
rectangular.

I'm not a woodworker so I don't know how this is would generally be done(I
imagine it would be extremely easy with a cnc). Obviously one can use a
chisel but I'm not proficient enough to make it look decent and probably
will screw up the wood.

I've seen forstner bits and counterbore bits but they all seem to have
create pilot hole.