Cook the green beans - Blanch green beans in boiling, salted water until crisp-tender, then rinse them under cold water to stop the cooking and preserve the bright green color. Set aside.
Combine and cook - Tear the mushrooms into bite-sized pieces and add them to a skillet. Season with salt and cook, stirring occasionally, until softened. Then reduce heat to medium and add the butter. Once the butter is melted and the mushrooms are coated, add the green beans, and 1 t salt. Cook, tossing often, until the butter begins to become brown and nutty. Add vinegar and a generous amount of pepper and cook a minute longer.
Finish and serve - Transfer the green beans and mushrooms to a platter and pour any pan sauce over the dish evenly. Garnish with crispy shallots and serve.