Heat a large skillet over medium-high heat. Add 2 tablespoons of the oil to the skillet and heat for another 30 seconds. Add the mushrooms, garlic and 3/4 teaspoon salt to the skillet and stir. Cook undisturbed, until the bottom layer of mushrooms is golden brown, about 4 minutes. Stir and repeat until the mushrooms are evenly browned and cooked through, about 10 minutes. Add the remaining 1 tablespoon oil and the shallots and cook, stirring often, about 2 minutes. Deglaze with the wine and simmer until almost reduced, about 1 minute. Add the chicken broth and butter and simmer to emulsify and thicken slightly, about 3 minutes.
Meanwhile, bring a large pot of water to a boil over high heat. Season generously with salt. Cook the pasta for 2 minutes less than the package instructions, until not quite al dente, about 10 minutes.
Using a slotted spoon, spoon the pasta directly from the water into the sauce over medium heat. Add 1/3 cup pasta water to the skillet. Sprinkle the bare pasta with 1 cup of the Parmesan. Toss well to combine and coat in the sauce. Simmer, tossing often, to finish the pasta, about 2 minutes. Stir in the arugula until just wilted and add the remaining Parmesan. Sprinkle with the pistachios. Top with more Parmesan before serving, if desired.