2 tablespoons extra-virgin olive oil, plus more for brushing
2 tablespoons balsamic vinegar
2 tablespoons chopped parsley
1 teaspoon minced rosemary
1 teaspoon minced thyme
Trim the stems from the artichokes and snap off the tough leaves. Cut off the top third of each artichoke and snip off the leaf tips with scissors. Steam the artichokes over moderate heat until tender when pierced at the bottom, about 30 minutes; check the water level halfway through cooking. Transfer to a plate.
Add the Jerusalem artichokes and potato to the steamer and steam until tender, about 40 minutes; again, check the water level halfway through.
Pull the center leaves from the artichokes and scrape out the hairy chokes with a small spoon. Set the 4 prettiest artichokes aside. Remove the leaves from the 2 remaining artichokes and refrigerate for another use.
Quarter the 2 artichoke bottoms and place in a bowl. Add the Jerusalem artichokes and potato and mash the vegetables to a smooth puree with a potato masher. Stir in 1 cup of the Parmesan and season with salt and pepper.
Preheat the oven to 350°. Spoon the stuffing into the centers of the 4 reserved artichokes. Separate the leaves of the artichokes slightly and use a small spoon to pack the remaining stuffing in between them. Set the artichokes on an oiled baking sheet, brush with olive oil and sprinkle each with 2 tablespoons of the remaining Parmesan cheese. Bake for about 30 minutes or until very hot and browned around the edges.
In a bowl, whisk the 2 tablespoons olive oil with the vinegar and season with salt and pepper. Set the stuffed artichokes on plates, drizzle on the vinaigrette and sprinkle with the parsley, rosemary and thyme. Serve hot.