Place the porcini mushrooms in a small bowl, add ¾ cup boiling water, and let sit 30 minutes. Strain, coarsely chop the mushrooms, and set aside. Pour the liquid through a coffee filter or fine sieve to strain; set aside.
Place a wok or large skillet over high heat; add the olive oil. When hot, add the shallot and garlic; sauté 1 minute. Add the reserved porcini mushrooms, wild mushrooms, thyme, oregano, salt and pepper; cook, stirring constantly, until softened and lightly browned, about 6 minutes. Add the Madeira wine, reserved porcini soaking liquid, and broth; cook until the liquid is reduced by half, about 4 minutes. Remove from heat.
Meanwhile, bring a pot of water to a boil over high heat. Add the pasta and cook until al dente. Drain, add the pasta to the mushroom sauce with the spinach. Stir until combined and the spinach is wilted.