Run guttering over flat roof

Hi,
Currently at the side of our house there is a large utility extension from the garage extending 2-3m into the garden. At the moment the gutter drain pipe from the roof of the main property is running above the flat roof then is diverted along a drain pipe diagonally along the back external wall of the prerogative to a drain halfway along the property (very ugly). I wanted to change so the drain pipe goes to the drain at the back of the flat roof extension.
I have two options, can anyone advise which would be best/most professional.

1) Run guttering along the side of the extension and join it up to what is already at the rear of the extension so it then goes into the rear drain; or

2) Run a drain pipe over the roof of the extension to the rear guttering of the extension so again it goes out the same drain.

Hopefully you can picture the scenario?

Any advice would be appreciated.

Thanks

Dan