While the water is heating, in a large skillet over medium-high heat, heat the 1/4 cup of olive oil until shimmering. Add about half the mushrooms and cook, stirring, until they begin to wilt and make space in the pan; then add the remaining mushrooms. Cook, stirring often, until they give up their liquid; then increase the heat to boil the liquid away and caramelize the mushrooms, 8 to 10 minutes.
Reduce the heat to medium, add the butter and let it melt. Stir in the garlic, rosemary, salt and red pepper flakes just until fragrant, 30 seconds. Add the vegetable broth and simmer until reduced by half, about 3 minutes. Reduce the heat to its lowest setting and cover to keep warm.
Meanwhile, add the pappardelle to the boiling water, and cook until al dente, 1 to 2 minutes less than the package directions. When the pasta is ready, use tongs to transfer it to the sauce, reserving the pasta water. Drizzle with olive oil, sprinkle with the parsley and toss to coat, adding a little reserved pasta water if the pan seems dry. Remove the skillet from the heat, and sprinkle with the grated cheese. Toss and serve hot,
I used dried gourmet mushrooms that I re-hydrated.