It sounds like you propose a lean-to nearly as wide as the original structure. It seems to me this is going to look way out of proportion, as well as presenting some potential design and structural issues.
It sounds like the current roof is a 4:12 pitch with 12 foot side walls. Add a 24 foot wide lean-to to this, and the lean-to roof is going to start at 12 feet and end up at 4 feet, assuming you keep the same roof pitch as the main structure. You'd have to use a flattish pitch of 2 in 12 to have decent headroom throughout the lean-to, and this raises questions about roof leaks and snow loads typical of such flat roofs.
A 24 foot wide lean-to will double the loading on the side wall, assuming you scab the lean-to onto the existing building. Is the existing wall structure and foundation sufficient to support twice as much roof and roof loading? Or do you plan to add a separate foundation for the lean to? If the latter, how do you do this without disturbing the support for the existing building?
I've seen big, flat-roof lean-to additions on small barns before, and in my experience they look awkward and perform poorly. I think you'd do better to scale back the size of the lean-to or build a whole new structure.