It a gutter thats "built in" to the roof. It allows more elaborate cornice moulding because you dont have to hang a gutter on the edge of your roof. Todays contractors like to roll out a gutter, whack it in place and be done. Yankee gutters required more talent and craftsmanship. I found this link showing a detail:
You probably need to look for a carpenter, or someone who leans towards old building restoration. Maybe contact a local sheetmetal shop for references, since the gutter is lined, usually copper, they may know a guy who could handle it.