Preheat the grill to 350°F Line two grill-proof pans with aluminum foil.
Remove the stems from the mushroom caps and finely chop the stems. Place the caps in a single layer on the foil-lined pans, gill-side up. Cook the mushrooms on the hot grill with the lid closed for 10-15 minutes or until the caps begin to soften.
Meanwhile, heat the oil in a large skillet over medium heat. Sauté the onion until it begins to soften, about 3 minutes. Add the chopped mushroom stems and cook 2 minutes. Add the garlic and paprika, and cook 1 minute. Add the turkey and cook until it's no longer pink, about 3 minutes. Remove from heat. Stir in 3/4 cup chopped dried tomatoes. Add the Parmesan and a grind or two of freshly ground black pepper, mixing until well blended. Add salt, if needed.
Stuff the mushroom caps generously with the filling, and top with a thin slice of Gruyère. Return the mushrooms to the grill. Close the cover and cook for 10 to 15 minutes or until the cheese has melted and the mushrooms are tender. Sprinkle with remaining dried tomatoes. Serve immediately.