Cook green beans in a medium pot of boiling salted water until bright green, about 2 minutes. Beans should be cooked enough not to be crunchy. Drain and rinse green beans under cold water to stop them from cooking more and retain their color. Set aside.
Heat oil in a large skillet over medium-high. Cook shallots, stirring occasionally, until browned, frizzled, and crisp, 6–8 minutes. Using a slotted spoon, transfer to paper towels to drain, and immediately season with salt. Set aside for serving.
Add mushrooms to same skillet and toss to coat with oil. Cook, stirring occasionally, until softened, about 4 minutes. Season with salt and cook, stirring occasionally, until starting to brown, about 4 minutes. Reduce heat to medium and add butter, reserved green beans, and 1½ tsp. kosher salt. Cook, tossing often and spooning foaming butter over vegetables, until butter darkens slightly and smells nutty, about 5 minutes. Add vinegar and season generously with pepper. Cook, stirring often, 1 minute.
Transfer green beans and mushrooms to a platter and spoon any sauce in pan over. Top with Parmesan and reserved crispy shallots.