Heat to a slow sizzle, and cook for 6 minutes or more, stirring often, until the onions are soft, wilted, and shiny, without any browning. Pour all the mushrooms into the pan, and spread them out. Sprinkle in the remaining salt, drop in the herb bouquet, then toss briefly and cover the pan. Raise the heat a bit, and cook, covered, for about 3 minutes, shaking the pan now and then to sweat the mushrooms.
Uncover, and continue to cook over fairly high heat, stirring frequently, as the mushrooms shrink and the liquid evaporates, 5 minutes or more. When the pan is dry and the mushrooms begin to brown, clear a spot, drop in the tomato paste, and toast it, stirring, for a minute or so; then stir it into the mushrooms.
When everything is browning again and just starting to stick, pour the Marsala all over. Stir constantly as the wine thickens and evaporates. When the mushrooms again start sticking to the bottom, pour in the warm mushroom water and 2 cups of the hot stock.
Bring to an active boil, stirring up any caramelization on the pan bottom. Lower the heat to keep the sauce bubbling gently all over the surface and cover the pan. Cook for about 20 minutes, occasionally stirring and adding stock to keep the mushrooms nearly covered in liquid; expect to add 1/2 cup or so at a time. Adjust the heat to keep the simmering.
Bring 6 quarts of salted water to a rolling boil and add the rigatoni. Stir periodically and cook until al dente as per box instructions.
Uncover the pan, and cook for another 20 minutes, continue to simmer the ragù adding stock as needed. When the mushrooms are thoroughly tender and the sauce has just thickened, remove and discard the herb bouquet. Season with salt and freshly ground black pepper to taste.
Place 2 cups of the cooked sauce in a sauté pan large enough to hold the drained pasta. Drain the pasta and reserve 1 cup of the pasta cooking water. Add the drained pasta to the bubbling sauce in the sauté pan, toss well until all the pasta is coated well, adding as needed the reserved pasta cooking water. Sprinkle with 1/2 cup of grated Grana Padano and serve.