In a very large bowl, whisk together 2 Tbsp of the oil, the vinegar, shallot, garlic, rosemary and salt. Thoroughly coat the mushrooms in the mixture and leave to marinate for 20 minutes.

Preheat the broiler. Transfer mushrooms to a baking pan that has sides at least 3/4 inch high and is large enough to hold the mushrooms in a single layer. Broil the mushrooms, turning them once, until they are nicely browned and tender, about 15 to 20 minutes.

Transfer the mushrooms to a platter and keep warm. Place the baking pan on top of the stove and add the stock and simmer over medium-high heat, stirring constantly, until the stock is thickened and reduced to about 1/3 cup.

Remove from the heat, stirring the remaining 2 tsp of the procini oil and the parsley and the sauce over the mushrooms. Serve immediately.