In a large, heavy skillet heat 3 tablespoons olive oil over medium-high heat. Add the mushrooms, onions, and garlic. Cook, stirring frequently until the mushrooms are tender and the onions are golden. Transfer the mushroom mixture to a medium bowl and let the mixture cool slightly. Add the prosciutto, parsley, Parmesan, salt, and pepper and stir to combine.
Gently pull apart the leaves of the trimmed artichokes to slightly widen the spaces between the leaves. Stuff the mushroom mixture in the spaces between the artichoke leaves. Place the artichokes in a heavy, high-sided pan so they fit snugly enough to stay upright. Pour the wine and 1 cup of water into the bottom of the pan. Pour the 1/4 cup of olive oil over the top of the artichokes. Place the pan over medium-high heat. Cover the pan and bring to a simmer. Reduce the heat to low and simmer for 45 minutes, until the artichokes are tender. Remove from the pot and serve immediately.