Step 1

Cook green beans in a medium pot of boiling salted water until bright green, about 2 minutes. Drain and rinse green beans under cold water to stop them from cooking more. Set aside.

Step 2

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.

Step 3

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. Diamond Crystal or 1 tsp. Morton 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.

Step 4

Transfer green beans and mushrooms to a platter and spoon any sauce in pan over. Top with Parmesan and reserved crispy shallots.Do ahead: Green beans can be blanched 1 day ahead. Pat dry; transfer to an airtight container or resealable bag and chill.