Trim off any soil caked onto mushrooms. Trim and discard discolored stem ends and tough stems of shiitakes. Quickly immerse the mushrooms in water, swishing them around to release soil and insects, then lift from water and drain.
Cut large mushrooms into about 1-inch pieces; leave the small mushrooms whole.
Peel and coarsely chop sunchokes.
In a 5- to 6-quart pan over high heat, combine oil, butter, mushrooms, sunchokes, shallots, and garlic. Stir often until mushroom juices evaporate and the vegetables are browned, about 15 minutes.
Add rosemary, oregano, and sherry; stir until sherry evaporates, about 2 minutes.
Spoon vegetables into a serving dish. Add salt and pepper to taste.
Sunchokes are also called Jerusalem artichokes. If making up to 1 day ahead, cover and chill. To reheat, stir over medium heat until hot, about 5 minutes. Garnish with fresh rosemary or oregano sprigs.