How to etch photograph on 7 micron gold plate?

A. Hi Dale
The technology you are looking for is called Photolithography.
It is a many stage process and involves a considerable amount of kit and many production stages. Take a look at en.wikipedia.org/wiki/Photolithography
You have posed an additional problem in asking to etch gold. The chemicals required to achieve this commonly include cyanides and are unavailable to private users. If you go this route you also need to consider disposal of waste chemicals, specific first aid etc.
A simpler solution might be to etch the substrate and then gold plate.
I also have a concern that, due to the reflectivity of gold, you may be disappointed by the result unless you back fill the etched lines with possibly a black filler. The old Daguerreotype process produced a similar image on silver and was not very satisfactory.
One small point. With gold at today's prices, have you checked that you are really getting the 7 microns you are paying for?
Also if it is a plated deposit, it will not be equal thickness across a flat sheet.

Because etching cannot reproduce continuous tones, you will have to convert your photographic original to a half tone image, see https://en.wikipedia.org/wiki/Halftone

This was routine with the old letterpress printing but you might have a problem with digital originals. Perhaps high contrast pixelation would work.

Or you might try laser etching.