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.
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 lightly browned, about 5 minutes. Add the garlic, shallot 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. Stir in the oregano and season with salt and pepper. Top with the remaining 1/2 cup parmesan.