Soak the dried mushrooms in boiling hot water for 30 minutes. Drain and pat dry.
Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs; reserve 1 cup cooking water, then drain.
Meanwhile, heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the squash, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until golden and tender, about 5 minutes. Transfer to a plate and set aside.
Chop the mushrooms. Add the remaining 1 tablespoon olive oil, the mushrooms, 1/4 teaspoon salt and a few grinds of pepper to the skillet. Cook, stirring occasionally, until the water evaporates and the mushrooms are crisp around the edges, about 5 to 8 minutes.
Add the garlic, shallot, oregano and red pepper flakes. Cook, stirring, until the shallot softens, about 2 minutes.
Add the pasta, squash and 1/2 cup of the reserved pasta cooking water to the skillet. Cook, stirring, until heated through,1 to 2 minutes. Stir in 1/2 cup parmesan, then stir in enough of the remaining cooking water to loosen. Season with salt and pepper. Top with the remaining 1/2 cup parmesan.