Heat 2 tablespoons oil in a large skillet over medium-high. Add half of the mushrooms and 1 teaspoon salt; cook, stirring occasionally, until mushrooms are golden brown, 6 to 10 minutes. Add 1/4 cup wine; cook, stirring to release browned bits from bottom of skillet, until liquid is completely evaporated, about 1 minute. Transfer cooked mushrooms to a plate. Repeat process with 2 tablespoons oil, remaining half of mushrooms, remaining 1 teaspoon salt, and remaining 1/4 cup wine. Wash and dry skillet; set aside.
Meanwhile, bring a large pot of salted water to a boil over medium-high. Add pasta, and cook according to package directions for al dente, 9 to 11 minutes. Reserving 2 cups pasta cooking liquid, then drain the pasta.
Heat garlic, butter, and remaining 1/4 cup oil in the large skillet over medium. Cook, stirring often, until garlic is just starting to brown, 2 to 4 minutes. Add shallots; cook, stirring often, until shallots are softened and garlic is golden, 4 to 6 minutes. Add rosemary; cook, stirring constantly, for 30 seconds. Reduce heat to medium-low. Stir in heavy cream, miso, Dijon, and 1/2 cup reserved pasta cooking liquid until evenly combined.
Add cooked pasta, mushroom mixture, and 1/2 cup reserved pasta cooking liquid to skillet with miso mixture. Cook, tossing constantly, until sauce evenly coats pasta, about 2 minutes, adding additional pasta cooking liquid, 1/4 cup at a time, as needed to thin sauce. Garnish with Parmesan, and serve with lemon wedges.
