You certainly have the height for a side shed roof off the gambrel and in rural areas, that is quite common on barns. Here is an idea, you could also T off one side of the existing shed. It could be a conventional roof design (dormer) that ties into the roof of your existing shed. You would have to stick build this yourself.
You could have the carriage doors on the side of the addition away from the existing shed on the dormer end or on the same side as the existing doors. This way you only have to build three sides and a roof. Once complete, you could cut a door in the side of the existing shed to tie the two together and put in a step or two between the elevations.
Well, the wife and I have discussed the options on a shed. We looked at it from all angles and have come up with a solution that removes all the questions I had about the type, style and size.
We're remodelling the bathroom instead.
By the time you finish that project, you will wish you had done the shed. It would be cheaper and easier, but bathrooms are important too.
Domestic tranquility and all that.